Download Young, Old, Conservative, and Bold: The Implications of

Young, Old, Conservative, and Bold: The Implications
of Heterogeneity and Finite Lives for Asset Pricing
Nicolae Gârleanu∗
Stavros Panageas†
Current Draft: July 2012
Abstract
We study the implications of preference heterogeneity for asset pricing. We use
recursive preferences in order to separate heterogeneity in risk aversion from heterogeneity in the intertemporal elasticity of substitution, and an overlapping-generations
framework to obtain a non-degenerate stationary equilibrium. We solve the model explicitly up to the solutions of ordinary differential equations, and highlight the effects
of overlapping generations and each dimension of preference heterogeneity on the market price of risk, interest rates, and the volatility of stock returns. We find that the
model has the potential to address some well-documented asset-pricing puzzles, while
also being consistent with certain salient facts about cross-sectional consumption and
wealth inequality.
∗
UC Berkeley, Haas School of Business, Centre of Economic Policy Research (CEPR), and National
Bureau of Economic Research (NBER), email: [email protected].
†
University of Chicago, Booth School of Business and National Bureau of Economic Research (NBER),
email: [email protected]. We would like to thank Andy Abel, Hengjie Ai, Nicole Branger,
John Cochrane, Mike Gallmeyer, John Heaton, Leonid Kogan, Debbie Lucas, Monika Piazzesi, Kathy Yuan,
four anonymous referees and participants at seminars and presentations at Berkeley Haas, the 2008 CARESSCowles Conference, Chicago Booth, Drexel, the 2007 Duke UNC Asset Pricing Conference, the 2011 Finance
UC Conference, HEC Paris, the 2012 ITAM Finance Conference, the 2007 NBER Summer Institute (AP),
Princeton University, the 2010 Royal Economic Society, the 2007 SAET conference, the 2008 SED conference, Stanford GSB, Studienzentrum Gerzensee, the 2008 UBC Winter Finance Conference, University of
Melbourne, University of Tokyo, and the 2008 Utah Winter Finance Conference for useful comments.
1
Macro-finance models routinely postulate economies populated by a “representative agent”
with constant relative risk aversion and intertemporal elasticity of substitution (IES). A legitimate question is how preference heterogeneity along both these dimensions, a widely
documented and intuitively plausible feature of reality, affects the conclusions of this parsimonious setup. We answer this question by identifying novel qualitative implications of such
heterogeneity and showing analytically and numerically its ability to match asset-pricing as
well as cross-sectional consumption and wealth data.
We study a continuous-time, overlapping-generations (OLG) economy (Blanchard (1985))
populated by agents with heterogeneous, recursive preferences (Kreps and Porteus (1978),
Epstein and Zin (1989), and Weil (1989)). The main differences between our setup and the
existing literature on preference heterogeneity are (i) OLG and (ii) recursive preferences.
The choice of these features is motivated by two considerations.
First, the OLG structure rules out the degenerate long-run wealth distributions obtaining
in models with infinitely-lived heterogeneous agents. Specifically, in such models it is quite
common that one type of agents dominates the economy in the long run, since preference
heterogeneity translates into persistent cross-sectional differences in average growth rates of
wealth. An immediate consequence is that all asset-pricing quantities (price-dividend ratios,
equity premia, etc.) converge to the constant levels obtaining when the economy is populated
by only one type of agent. As a result, it becomes difficult to compare these models to a
large body of empirical literature predicated on a non-degenerate stationary distribution
for these quantities.1 By contrast, our framework generates non-degenerate cross-sectional
distributions of wealth and consumption, through the combination of finite lifetimes and
the constant arrival of newly born agents endowed with the same labor earnings. As a side
benefit, the OLG structure enables an empirically disciplined link between the model’s crosssectional wealth and consumption distributions and the paths of life-cycle earnings observed
in the data.
1
To provide an example, the empirical literature on return predictability considers regressions of excess
returns on a constant and various predictors, such as the price-to-dividend ratio. The identifying assumption
of ordinary least squares (in large samples) requires that the regressors have non-degenerate asymptotic
distributions, i.e., do not converge to constants (otherwise, the variance-covariance matrix is singular).
2
Second, by using recursive rather than time-additive preferences we can separate — both
conceptually and quantitatively — the effects of risk-aversion and IES heterogeneity, and
study their interaction.
We show that these features of the model — OLG and heterogeneity of recursive preferences — help address a number of stylized facts in asset pricing, which can be grouped
in three broad categories, concerning (i) the market price of risk, (ii) the interest rate, and
(iii) the return volatility and equity premium. We discuss each of them in turn.
The market price of risk, defined as the ratio of stock-market excess return to stockmarket volatility, can increase above the levels that would obtain with heterogeneous CRRA
utilities. To be specific, in the special case of heterogeneous CRRA utilities our OLG structure preserves a familiar result in the literature: The market price of risk is negatively correlated with the underlying aggregate-consumption shock and given by a weighted average of
the market prices of risk in two representative-agent economies, populated exclusively by the
more risk-averse, respectively the less risk-averse, agents of our model. Novel results obtain
when agents have heterogeneous recursive preferences. The optimal consumption-allocation
rule endogenously generates persistence in individual agents’ consumption growth, although
aggregate-consumption growth is i.i.d.. Except in the special CRRA case, agents’ marginal
rates of substitution are affected by such persistence, and so is the market price of risk.
We provide conditions under which the resulting market price of risk exceeds the one under
CRRA preferences (keeping risk-aversion coefficients fixed). Interestingly, in some illustrative examples the market price of risk exceeds even the value obtaining in the economy
populated exclusively by the agent with the higher risk-aversion coefficient.
Our second set of results concerns interest rates. We show that even if all agents have
homogeneous preferences, interest rates in an OLG economy are lower than in the respective
infinitely-lived agent economy for the low values of IES that generate the risk-free puzzle in
such an economy (Weil (1989)). The reason is that finitely-lived agents faced with realistic
life-cycle earnings profiles need to save in order to provide for themselves in old age. These
increased savings lower the interest rate.
3
Our third set of results concerns the volatility of stock-market returns and the equity
premium. We first show that, if agents have different IES, but the same risk-aversion coefficient, then the volatility of the stock market and the equity premium (as well as the market
price of risk) are constant and equal to their respective values in an economy populated by a
single agent with the same risk aversion. In our model, therefore, risk-aversion heterogeneity
is essential for addressing certain empirical asset-pricing properties.
However, for a given degree of risk-aversion heterogeneity, heterogeneity in the IES can
affect the results significantly. Under certain conditions, we show that the primary effect
of IES heterogeneity is to control variability in the real interest rate, while the primary
effect of risk aversion heterogeneity is to control time variation in the market price of risk.
Accordingly, the two dimensions of heterogeneity can be combined to produce discount-rate
variability that is negatively related to the underlying aggregate-consumption shock and is
predominantly driven by the variability of the equity premium, rather than that of the real
interest rate.
Quantitatively, we find that — for realistically calibrated life-cycle earnings profiles and
an appropriate choice of the joint distribution of IES and risk aversion across agents —
we can obtain a small and non-volatile interest rate along with a substantially larger and
more variable equity premium. A novel feature of our paper is that the same assumptions
on preference specifications that are required to reproduce asset-pricing facts also imply
cross-sectional consumption and wealth populations that share many salient features of the
data. In particular, the model can simultaneously produce large variation in asset prices and
relatively low variation in measures of cross-sectional consumption and wealth dispersions.2
Our paper relates primarily to the analytical asset-pricing literature on preference heterogeneity.3 As already mentioned, this literature does not separate IES heterogeneity from
risk-aversion heterogeneity. Furthermore, with the notable exception4 of Chan and Kogan
2
Constantinides and Duffie (1996) show that their heterogeneous-income model can generate an arbitrary
stochastic discount factor. However, this may require a large degree of time variation in cross-sectional
consumption dispersion.
3
Contributions include Dumas (1989), Wang (1996), Chan and Kogan (2002), Bhamra and Uppal (2009),
Longstaff and Wang (2008), and Zapatero and Xiouros (2010).
4
See also Zapatero and Xiouros (2010) for a discrete-time version of Chan and Kogan (2002).
4
(2002), these models imply generically non-stationary dynamics. The key assumption that
ensures stationarity in Chan and Kogan (2002) is that an agent’s utility derives exclusively
from her consumption relative to aggregate consumption. We use instead an OLG approach
to obtain stationarity (see, e.g.,5 Spear and Srivastava (1986)), which allows us to study
standard CRRA and recursive-utility specifications. In doing so, we can better isolate the
effects of preference heterogeneity, while at a practical level we can avoid the large volatility
of the real interest rate that is associated with (multiplicative) external habit formation.
A theoretical literature has considered preference aggregation for infinitely-lived agents
with heterogeneous, recursive preferences. Early contributions studied deterministic environments.6 Subsequently, other authors investigated the existence of equilibrium and recursive
algorithms for the construction of a representative agent in stochastic environments.7 From
a methodological perspective, we develop a novel approach to determining equilibrium allocations, which is distinct from the central-planning approaches developed in Kan (1995)
and Dumas et al. (2000). Our alternative is better suited for an OLG setup, since the OLG
feature precludes usage of the existing central-planning approaches. A useful byproduct
of our approach is that we can provide analytic expressions for the equilibrium prices and
quantities up to the solution of a system of ordinary differential equations. Besides allowing
an accurate and efficient computation of the equilibrium, these expressions make it possible to characterize the new equilibrium properties associated with recursive preferences (as
compared with CRRA preferences).8
Unlike the case of infinitely-lived agents, in which survival of all agent types is parameter dependent,9 in our OLG framework survival obtains generically. Accordingly, we can
analyze the effect of parameter constellations (e.g., the heterogeneous CRRA case) that are
incompatible with stationarity when agents are infinitely lived. Additionally, by analyzing
5
Spear and Srivastava (1986) discusses the existence of stationary Markov equilibria in OLG models.
However, it focuses on heterogeneous goods (rather than preferences), and the Markov state space contains
past prices rather than the consumption distribution.
6
See, e.g., Lucas and Stokey (1984) and Epstein (1987).
7
See, e.g., Ma (1993), Kan (1995), Duffie et al. (1994), and Dumas et al. (2000).
8
Backus et al. (2008) give a closed-form solution for the special case where one type of agents is risk
neutral and has zero IES, whereas the other type of agents is infinitely risk averse and has infinite IES.
9
See the recent contribution of Branger et al. (2012).
5
equilibrium quantities around the steady state, we can obtain a more detailed characterization of asset-pricing quantities than the previous literature.
Some of our model’s channels are reminiscent of leading representative-agent models.
Similar to Campbell and Cochrane (1999), the market price of risk increases following negative shocks. In Campbell and Cochrane (1999) this property is due to the assumed sensitivity
to shocks of the representative-agent risk aversion, while in our model it arises as a result
of the endogenous redistribution of wealth from the less to the more risk averse agents in
bad times.10 Also, similar to Bansal and Yaron (2004), persistent consumption growth coupled with recursive preferences generate a higher market price of risk. However, in contrast
to Bansal and Yaron (2004), this persistence is the endogenous result of the consumptionallocation rule and it pertains to the consumption growth of individual agents. It obtains
even if aggregate consumption growth is i.i.d.
Another relevant literature employs numerical methods to solve life-cycle-of-earnings
models in general equilibrium. The paper most closely related to ours is Gomes and Michaelides
(2008).11 Gomes and Michaelides (2008) obtains joint implications for asset returns and
stock-market participation decisions in a rich setup that includes costly participation, heterogeneity in both preferences and income, and realistic life cycles. This paper also assumes,
however, that the volatility of output, the volatility of stock market returns, and the equity
premium are driven by exogenous, random capital-depreciation shocks, rather than the commonly assumed total-factor-productivity shocks. This assumption implies that stock-market
volatility is due almost entirely to exogenous variation in the quantity of capital, rather than
to fluctuations in the price of capital (Tobin’s q), in contrast to the data where most of the
return volatility is due to fluctuations in the price of capital. This exogenous source of return
10
This result was first shown for CRRA utilities by Dumas (1989), and emphasized by Chan and Kogan
(2002). We also note that a large asset-pricing literature obtains time-varying market prices of risk as a
result of the coexistence of multiple goods or production factors. Indicative examples include Piazzesi et al.
(2007), Santos and Veronesi (2006), Tuzel (2010), Papanikolaou (2010), and Gomes et al. (2009). Our model
differs in that the result is not due to assumptions on relative endowments of different goods and production
factors, but to the interaction of agents with different preferences.
11
Other related work includes Storesletten et al. (2007) and Guvenen (2009). Storesletten et al. (2007)
study an OLG economy with homogeneous preferences and uninsurable income shocks. Similar to the OLG
model of Gomes and Michaelides (2008), Guvenen (2009) studies preference heterogeneity and limited market
participation, but his model features inifinitely lived agents.
6
volatility is essential for their model to produce a non-negligible equity premium.12
We consider exclusively endowment shocks in a Lucas (1978)-style economy. While this
more conventional asset-pricing framework abstracts from modeling investment, it has the
advantage that stock-market fluctuations are due to endogenous variations in the price of
capital. It also allows us to readily compare our results to the large asset-pricing literature
using such a framework. More importantly, endogenizing the price of capital generates new
insights compared to Gomes and Michaelides (2008). For instance, even though our theoretical results on the market price of risk support the conclusions of Gomes and Michaelides
(2008), we also find that some of the joint distributions of intertemporal elasticity of substitution and risk aversion discussed in Gomes and Michaelides (2008) may increase the market
price of risk at the cost of attenuating the volatility of returns and the equity premium.
Another difference with the calibration-oriented literature is the tractability of our framework, which allows us to obtain analytical results. Therefore, our work is complementary to
quantitative exercises that feature richer setups, but must sacrifice partly the transparency
of the mechanisms involved.
We also relate to the literature that analyzes the role of OLG in asset pricing.13 Many
of these models combine the OLG structure with other frictions or shocks to drive incomplete risk sharing across generations, so that consumption risk is disproportionately high for
cohorts participating predominantly in asset markets. Even though we think these channels
important for asset pricing, we do not include them in order to isolate the intuitions pertaining to preference heterogeneity. Another source of difference is that we model births and
deaths in continuous time, similar to Blanchard (1985). As a result, our model produces
implications for returns over any duration (month, year, etc.), and not just over the lifespan
of a generation.14
The paper is structured as follows. Section 1 presents the model. Section 2 discusses
12
As Gomes and Michaelides (2008) acknowledge, “Without those shocks, our economy would still have a
high market price of risk, but a negligible equity premium.”
13
Examples of such papers are Abel (2003), Storesletten et al. (2007), Constantinides et al. (2002), Farmer
(2002), Piazzesi and Schneider (2009), Heaton and Lucas (2000), and Gârleanu et al. (2009).
14
Farmer (2002) analyzes a stochastic version of Blanchard (1985) in discrete time, but all agents maximize
the same logarithmic preferences.
7
the solution obtaining with preference homogeneity, which allows us to isolate the effects
of overlapping generations. Section 3 introduces preference heterogeneity, and develops the
analytical implications of the model for asset pricing. Section 4 discusses quantitative implications and Section 5 concludes. All proofs are in the appendix.
1
Model
1.1
Demographics and preferences
Our specification of demographics follows Blanchard (1985) and Yaari (1965). Time is continuous. Each agent faces a constant hazard rate of death π > 0 throughout her life, so
that a fraction π of the population perishes per unit of time. Simultaneously, a cohort of
mass π is born per unit of time. Given these assumptions, the time-t size of a cohort of
agents born at some time s < t is given by πe−π(t−s) ds, and the total population size is
Rt
πe−π(t−s) ds = 1.15
−∞
To allow for the separation of the effects of the IES and the risk aversion, we assume
that agents have the type of recursive preferences proposed by Kreps and Porteus (1978),
Epstein and Zin (1989), and Weil (1989) in discrete-time settings and extended by Duffie
and Epstein (1992) to continuous-time settings. Specifically, an agent of type i maximizes
Vsi
∞
Z
f
= Es
ciu , Vui
du ,
(1)
s
where f i (c, V ) is given by

f i (c, V ) ≡
1 
αi
c
((1 −

αi
γ i) V
)
αi
−1
1−γ i
− (ρ + π) 1 − γ i V  .
(2)
The function f i (c, V ) aggregates the utility arising from current consumption c and the value
−1
function V. The parameter γ i > 0 controls the risk aversion of agent i, while (1 − αi )
15
gives
We assume no population growth for simplicity. Introducing population growth is a straightforward
extension.
8
the agent’s IES. We assume that αi < 1, so that the IES ranges between zero and infinity.
The parameter ρ is the agent’s subjective discount factor. The online appendix16 gives a
short derivation of the objective function (1) as the continuous-time limit of a discrete-time,
recursive-preference specification with random times of death.
To study the effects of heterogeneity in the most parsimonious way, we assume that there
are two groups of agents labeled A and B: agents in group A have risk aversion γ A and
−1
−1
. At
, whereas agents in group B have risk aversion γ B and IES 1 − αB
IES 1 − αA
every point in time a proportion υ A ∈ (0, 1) of newly born agents are of type A, while the
remainder υ B = 1 − υ A are of type B. For the rest of the paper we maintain the convention
γ A ≤ γ B , and use superscripts to denote the type i ∈ {A, B} of an agent.
1.2
Endowments, earnings, and dividends
Aggregate output in the economy is given by
dYt
= µY dt + σY dBt ,
Yt
(3)
where µY and σY are constant parameters and Bt is a standard Brownian motion.17
At time t, an agent born at time s is endowed with earnings yt,s , where
yt,s = ωYt [G (t − s)] ,
ω ∈ (0, 1).
(4)
G (t − s) ≥ 0 is a function of age that controls the life-cycle earnings profile. For some results
in the paper we can consider an arbitrary function G (t − s). For tractability, some results
use the parametric form
G (u) = B1 e−δ1 u + B2 e−δ2 u .
(5)
This parametric form is flexible enough to reproduce the hump-shaped pattern of earnings
16
17
Available at http://faculty.chicagobooth.edu/stavros.panageas/research/OLGextapp.pdf.
We fix throughout a probability space and a filtration generated by B satisfying the usual conditions.
9
observed in the data, as we illustrate in the next section.
Aggregate earnings are given by
Z
t
−π(t−s)
πe
Z
t
πe
yt,s ds = ωYt
Z
G (t − s) ds = ωYt
−∞
−∞
We assume throughout that
−π(t−s)
R∞
0
∞
πe−πu G (u) du.
(6)
0
πe−πu G (u) du < ∞, and normalize the value of this integral
to 1 by scaling G. The aggregate earnings are therefore given by ωYt , the remaining fraction
1 − ω of output Yt being paid out as dividends Dt ≡ (1 − ω) Yt by the representative firm.
Following Lucas (1978), we assume that the representative firm is a “Lucas tree”, i.e., it
simply pays dividends without facing any economic decisions.
1.3
Markets and budget constraints
Agents can allocate their portfolios between shares of the representative firm and instantaneously maturing riskless bonds, which pay an interest rate rt per dollar invested. The
supply of shares of the firm is normalized to one, while bonds are in zero net supply. The
price St of each share evolves according to
dSt = (µt St − Dt ) dt + σt St dBt ,
(7)
where the coefficients µt and σt are determined in equilibrium, as is the interest rate rt .
Finally, agents can access a market for annuities through competitive insurance companies
as in Blanchard (1985). Specifically, an agent born at time s who owns financial wealth equal
to Wt,s at time t can enter a contract with an annuity company that entitles the agent to
receive an income stream of πWt,s per unit of time. In exchange, the insurance company
collects the agent’s financial wealth when she dies. Entering such an annuity contract is
optimal for all living agents, given that they have no bequest motives. The law of large
numbers implies that insurance companies collect πWt per unit of time from perishing agents,
where Wt denotes aggregate financial wealth. This allows them to pay an income flow of
πWt to survivors and break even.
10
Letting θt,s denote the dollar amount invested in shares of the representative firm,18 an
agent’s financial wealth Wt,s evolves according to
dWt,s = (rt Wt,s + θt,s (µt − rt ) + yt,s + πWt,s − ct,s ) dt + θt,s σt dBt ,
1.4
Ws,s = 0.
(8)
Equilibrium
The definition of equilibrium is standard. An equilibrium is given by a set of adapted
processes {ct,s , θt,s , rt , µt , σt } such that (i) the processes ct,s and θt,s maximize an agent’s objective (1) subject to the dynamic budget constraint (8), and (ii) markets for goods clear, i.e.,
Rt
Rt
−π(t−s)
πe
c
ds
=
Y
,
and
markets
for
stocks
and
bonds
clear
as
well:
πe−π(t−s) θt,s ds =
t,s
t
−∞
−∞
Rt
St and −∞ πe−π(t−s) (Wt,s − θt,s ) ds = 0.
2
Homogeneous preferences
Our model setup has two main features: OLG and preference heterogeneity. In preparation
for the main results of the paper, this section clarifies the asset-pricing implications of OLG
(absent preference heterogeneity).
Proposition 1 Suppose that all agents have the same preferences αA = αB = α, γ A = γ B = γ ,
and consider the following non-linear equation for r:
r = ρ + (1 − α) (µY + π (1 − β)) − γ (2 − α)
σY2
,
2
(9)
where
Z
β=ω
∞
G(u)e
2 −µ )u
−(r+π+γσY
Y
du
π+
0
ρ
α γ 2
−
r + σY
.
1−α 1−α
2
(10)
Suppose that r̄ is a root of (9) and r̄ > µY − γσY2 .19 Then there exists an equilibrium where
18
We assume standard square integrability and transversality conditions. See, e.g., Karatzas and Shreve
(1998) for details.
19
Lemma 2 contains sufficient — but not necessary — conditions on the parameters for the existence of
11
the interest rate, the expected return, and the volatility of the stock market are all constant
and given, respectively, by rt = r̄, µt = r + γσY2 , and σt = σY .
Proposition 1 shows that in the absence of preference heterogeneity the equity premium
and the volatility of the stock market in our OLG economy are identical to the respective
quantities in a standard, infinitely-lived representative-agent model with the same preferences. (See, e.g., Weil (1989).) Accordingly, the OLG feature does not help address issues
such as the equity premium, the excess volatility puzzle, the predictability of returns, etc..
The only interesting asset-pricing implication of the OLG assumption is that the interest rate
differs from the respective infinitely-lived representative-agent economy: in the infinitelylived representative-agent model the interest rate is given by r = ρ+(1 − α) µY −γ (2 − α)
2
σY
2
,
while equation (9) contains the additional term (1 − α) π (1 − β) .
The source of the difference is that in an OLG economy only the Euler equation (and
hence the per-capita consumption growth) of existing agents matters. The per-capita consumption growth of existing agents is in general different from the growth rate of aggregate
consumption, because of deaths and births. Absent births, the death of a fraction π of the
population per unit of time would imply a per-capita consumption growth rate of µY + π.
In the presence of births, however, a portion of aggregate consumption accrues to arriving agents. Collectively, these agents consume πct,t , which is a fraction
πct,t
Ct
of aggregate
consumption. Therefore, the combined effect of births and deaths is that the per capita
consumption growth of existing agents is µY + π 1 − cCt,tt .
As we show in the appendix, when all agents have homogeneous preferences,
ct,t
Ct
is con-
stant and equal to the constant β given in Proposition 1. In the appendix (Lemma 2), we
also extend some results of Blanchard (1985) and show that under certain conditions β > 1
and, consequently, the interest rate of our economy (equation (9)) is lower than the interest
rate in the respective economy featuring an infinitely-lived agent.
Figure 1 depicts the quantitative magnitude of the difference in riskless rates between
an economy featuring an infinitely lived agent and our OLG economy. To expedite the
such a root.
12
γ=4
γ=10
0.04
0.07
0.035
0.06
0.03
0.05
0.025
0.04
Inf. lived Agent
r
r
Inf. lived Agent
0.02
OLG
0.03
0.015
0.02
0.01
0.2
OLG
0.01
0.3
0.4
0.5
0.6
0.005
0.2
IES
0.3
0.4
0.5
0.6
IES
Figure 1: Interest rates when all agents have identical preferences. The dashed line pertains to an
economy populated by an infinitely-lived agent, while the other two lines to our OLG model. The
continuous line obtains when using the exact path for G(u) reported in Hubbard et al. (1994) and
performing the integration in equation (10) numerically. The dash-dot line obtains when using nonlinear least squares to project G (·) on a sum of scaled exponential functions, and then calculating
the integral in equation (10) exactly. For the computations, we set a positive, but small, value of
ρ = 0.001 to illustrate that the high risk free rates (in the case where agents are infinitely lived) are
not the result of a high discount rate. We set µY = 0.02 and σY = 0.041 to match the mean and
the volatility of the yearly, time-integrated consumption growth rate, π = 0.02 to match the birth
rate, and ω = 0.92 to match the share of labor income in national income. Details on the sources
of the data and further discussion of these parameters are contained in the appendix (Section B).
presentation of the main results of our paper, we refer the reader to the appendix (Section
B) for a detailed justification of the parameters that we use for calibration. In short, we
choose µY and σY to reproduce average, annual, time-integrated consumption growth and its
volatility, π to reproduce the average birth rate in the economy, and ω to capture the labor
share of national income. For G(t−s) we use the exact path of life-cycle earnings reported in
Hubbard et al. (1994), which we depict in Figure 2. (For future reference, it is useful to note
that interest rates remain basically unchanged when we use the parametric specification for
G(t − s) of equation (5) with parameters estimated by non-linear least squares.) Figure 1
13
2
1.8
Exact
1.6
1.4
G(t−s)
1.2
1
0.8
Sum of exponentials
0.6
0.4
0.2
0
20
30
40
50
60
70
80
90
100
110
120
Age
Figure 2: Hump-shaped profile of earnings over the life-cycle. The continuous line reports the
profile estimated by Hubbard et al. (1994), while the broken line depicts the non-linear leastsquares projection of the earnings profile in the data on a sum of scaled exponentials G (u) =
B1 e−δ1 u + B2 e−δ2 u . The estimated coefficients are B1 = 30.72, B2 = −30.29, δ1 = 0.0525, and
δ2 = 0.0611.
shows that, for the low levels of IES that lead to implausibly high interest rates in the
economy featuring an infinitely-lived agent, the OLG counterpart generates lower interest
rates.
We summarize the role of the OLG feature of our model as follows: Besides helping lower
the equilibrium interest rate, the OLG feature adds little in terms of resolving asset-pricing
puzzles such as the equity premium, predictability, and excess volatility. However, as we
show shortly, it plays a key role in terms of ensuring stationary consumption and wealth
distributions with preference heterogeneity.
3
Heterogeneous preferences
We can now turn to the main results of the paper, by allowing agents to be heterogeneous
(γ A 6= γ B , αA 6= αB ).
14
To motivate our construction of the equilibrium and obtain some basic implications of
heterogeneity, we derive a few results for heterogeneous CRRA preferences (1 − αi = γ i ,
γ A < γ B ). We start with two definitions. First, we let Xt denote the fraction of aggregate
output consumed by type-A agents:
Xt ≡
υAπ
Rt
−∞
e−π(t−s) cA
t,s ds
.
Yt
(11)
Second, we define the “market price of risk” (or Sharpe ratio) as
κt ≡
µt − rt
.
σt
(12)
Since agents can dynamically trade in a stock and a bond, they face dynamically complete
markets over their lifetimes. As a result, letting ξt denote the stochastic discount factor,
their consumption processes satisfy the relation
e−(ρ+π)(t−s)
cit,s
cis,s
−γ i
= e−π(t−s)
ξt
, for i ∈ A, B.
ξs
(13)
Using (13) to solve for cit,s and substituting into the definition of Xt gives
A
Xt ≡
−
υ πξt
1
γA
Rt
e
−∞
−π−
ρ
γA
1
(t−s) A γ A
cs,s ξs ds
Yt
.
(14)
Letting σX denote the diffusion coefficient of Xt , applying Ito’s Lemma to (14) and isolating
diffusion coefficients yields
σX =
κt
− σY
γA
Xt ,
(15)
where we have used (3) and the fact that the stochastic discount factor evolves as
dξt
ξt
=
−rt dt − κt dBt . Performing a similar computation to determine the diffusion coefficient of
the consumption share of agent B and noting that the consumption share of agent B is equal
15
to (1 − Xt ) yields
−σX =
κt
− σY
γB
(1 − Xt ).
(16)
Adding equations (15) and (16) and solving for κt yields
κt = Γ (Xt ) σY ,
(17)
where
Γ (Xt ) ≡
Xt 1 − Xt
+
γA
γB
−1
.
(18)
Equation (17) has the familiar form one encounters in single-agent setups:the market price
of risk is the product of the “representative agent’s” risk aversion Γ(Xt ) and the volatility of
aggregate consumption. Alternatively phrased, the market price of risk with heterogeneous
CRRA agents is identical to the market price of risk in an otherwise identical economy
populated by a single, fictitious, expected-utility-maximizing “representative” agent with
risk aversion given by Γ(Xt ).20
An important difference with single-agent setups, however, is that the risk aversion of
the representative agent is not constant, but rather time varying and negatively correlated
with aggregate-consumption growth. To see this fact, use (17) inside (15) to obtain
σX (Xt ) = Xt
Γ (Xt )
− 1 σY .
γA
(19)
Since Γ(Xt ) > γ A for Xt ∈ (0, 1), equation (19) implies σX > 0. Hence, positive innovations to aggregate consumption increase the consumption share of type-A agents. At the
same time, Γ (Xt ) is a declining function of Xt , so that whenever aggregate consumption
experiences a positive innovation, Γ (Xt ) declines.
20
Similar results to equation (17) have been established in the literature. See, e.g., Dumas (1989), Chan
and Kogan (2002), and Zapatero and Xiouros (2010).
16
The economic intuition behind this result is straightforward. Less risk-averse agents
(type-A agents) invest more heavily in stocks than more risk-averse agents (type-B agents).
As a result, a positive aggregate shock raises the wealth and consumption shares of less
risk-averse agents. When less risk-averse agents own a larger fraction of aggregate wealth,
the market price of risk is low, since these agents require a relatively smaller compensation
for holding risk. Conversely, a negative shock increases the consumption and wealth shares
of more risk averse agents, and consequently the market price of risk.
A mathematical implication of equations (17) and (19) is that both the Sharpe ratio κt
and the diffusion coefficient σX only depend on the level of Xt . This observation motivates
us to search for an equilibrium with the properties that the Sharpe ratio and the interest rate
depend exclusively on Xt , and moreover Xt is Markov (i.e., its drift and volatility depend
exclusively on Xt ). The next proposition describes such an equilibrium, while also allowing
for general, recursive preferences.
Proposition 2 Let Xt and G(t − s) be defined as in equations (11) and (5), respectively,
and let βti ≡
cit,t
,
Yt
i ∈ {A, B}, denote the consumption of a newly-born agent of type i as a
fraction of aggregate output. Finally, let Γ (Xt ) be defined as in (18), let Θ (Xt ), ω A (Xt ),
ω B (Xt ) , and ∆ (Xt ) be defined as
Xt
1 − Xt
+
,
A
1−α
1 − αB
Xt
ω B (Xt ) ≡ 1 − ω A (Xt ) ,
ω A (Xt ) ≡ A Γ(Xt ),
γ
A
B
γ +1
γ +1
∆ (Xt ) ≡ ω (Xt )
+ (1 − ω (Xt ))
,
γA
γB
Θ (Xt ) ≡
(20)
(21)
(22)
and assume functions g i (Xt ) and φj (Xt ), for i ∈ {A, B}, j ∈ {1, 2}, that solve the system
of ordinary differential equations (A.18) and (A.21) in the appendix. Then there exists an
equilibrium in which Xt is a Markov diffusion with dynamics dXt = µX (Xt )dt + σX (Xt )dBt
17
given by
Xt Γ (Xt ) − γ A
h
i
σX (Xt ) =
σY ,
Γ(Xt )
1−γ A −αA g A0
1−γ B −αB g B0
A
X
(1
−
X
)
−
+
γ
t
t
B
A
A
B
B
γ
α
g
α
g
r (Xt ) − ρ
µX (Xt ) = Xt
+ nA (Xt ) − π − µY + υ A πβ A (Xt ) − σY σX (Xt ) ,
1 − αA
(23)
(24)
with
g i0
σX (Xt ) ,
gi
i∈{A,B}





X
1
r (Xt ) = ρ +
µY − π 
υ i β i (Xt ) − 1

Θ (Xt ) 
κ (Xt ) = Γ (Xt ) σY +
X
i
ω (Xt )
1 − γ i − αi
αi
(25)
(26)
i∈{A,B}
1 Xt nA (Xt ) + (1 − Xt ) nB (Xt ) ,
−
Θ (Xt )
ni (Xt ) given by
2
2 − αi
αi + γ i − 1 g i0
2
n (Xt ) = i
κ (Xt ) +
σX (Xt )
2γ (1 − αi )
2γ i αi
gi
γ i − αi 1 − γ i αi + γ i − 1 g i0
−
σX (Xt ) κ (Xt ) ,
γ i (1 − γ i ) (1 − αi ) αi g i
i
(27)
and β i (Xt ) = g i (Xt )(φ1 (Xt ) + φ2 (Xt )) for i ∈ {A, B}.
Proposition 2 asserts that the consumption share of type-A agents (Xt ) follows a Markov
diffusion and that the market price of risk and the interest rate are exclusively functions
of Xt . In preparation for the discussion of the asset-pricing implications of the model, it is
useful to note some additional properties of Xt .
3.1
Consumption share Xt
A property of models featuring preference heterogeneity and infinitely lived agents is that in
many circumstances the consumption distribution eventually becomes degenerate:21 Different agents have persistently different mean growth rates of wealth and eventually one group
21
See, e.g., Dumas (1989) and Cvitanic and Malamud (2010).
18
dominates.
The OLG feature of our model, however, implies generically that no group of agents
becomes extinct in the long run. To establish this result, we start by noting that equation
(23) implies σX (0) = 0, while equation (24) gives µX (0) = υ A πβ A (0). Since the value of
life-time earnings of any newly born agent is strictly positive, so is any newly-born agent’s
consumption. Thus, β A (Xt ) =
cA
t,t
Yt
> 0 and, as a consequence, µX (0) > 0. Proceeding
in a similar fashion, equations (18) and (23) imply σX (1) = 0, and from (24) we deduce
µX (1) = −υ B πβ B (1) < 0.
An implication of the above facts is that the process for Xt does not get absorbed at
the points 0 or 1.22 Intuitively, although the mean growth rates of wealth differ across
agents, eventually all agents perish, regardless of their accumulated financial wealth. All
newly-born agents enter the economy with no financial wealth and with human capital that
is equal across preference groups and proportional to the level of output. As a result, each
group of agents receives a minimum inflow of new members whose consumption is a non-zero
fraction of aggregate output, ensuring that no type of agent dominates the economy.
Equations (23) and (24) also help illustrate the importance of risk-aversion heterogeneity
(as opposed to IES heterogeneity) in terms of obtaining non-trivial variation of the consumption distribution. Indeed, suppose that αA 6= αB and γ A = γ B , so that agents differ in their
IES, but not their risk aversion. Then equation (18) implies Γ (Xt ) = γ A = γ B for any value
of Xt , and therefore equation (23) implies σX = 0 for all Xt . The implications of this fact
are summarized in the following corollary to Proposition 2.
Corollary 1 Consider the setup of Proposition 2 and impose the restriction γ A = γ B = γ.
Then σX = 0 at all times, the market price of risk is given by the constant κ = γσ for all Xt ,
and there exists a steady state featuring a constant interest rate and a constant consumption
share of type-A agents given by some value X̄ ∈ (0, 1).
Corollary 1 states that when agents differ only in their IES, Xt converges deterministically
22
Technical details on boundary behavior and an analytic approach for the computation of the stationary
density (which we use when we derive the stationary means of the quantities in our quantitative analysis)
are given in Karlin and Taylor (1981).
19
3.1
IESA=1/γA, IESB=0.05
γA=2.5
γB=3
3.05
3
κ(Xt)/σY
maximum risk aversion
CRRA
2.95
2.9
2.85
0
0.02
0.04
0.06
0.08
0.1
Xt
0.14
0.16
Figure 3: An illustration of Propositions 2 and 3. The ratio
κ(Xt )
σY
aversion of even the most risk-averse agent.
0.12
0.18
0.2
may be higher than the risk
to a constant, and so does the interest rate. The market price of risk remains constant
throughout at the level γσ. In short, Corollary 1 asserts that risk-aversion heterogeneity is
essential for the model to have any interesting asset-pricing implications. Therefore, for the
rest of the paper we assume γ A < γ B .
3.2
Market price of risk (κt )
In the case where agents have heterogeneous CRRA preferences (1 − αi = γ i for i ∈ {A, B})
equations (25) and (23) become identical to equations (17) and (19), respectively. As we
discussed earlier, this implies that the Sharpe ratio in our economy is identical to the Sharpe
ratio of an economy populated by a representative agent featuring time-varying risk aversion.
Moreover, the risk aversion of that representative agent is a weighted average of the risk aversions of the individual agents, with weights that are functions of each agents’ consumption
shares.
A novel implication of equation (25) is that when agents have recursive preferences (1 −
20
t)
αi 6= γ i for some i ∈ {A, B}), the ratio κ(X
is no longer equal to Γ (Xt ) . Instead, it
σY
i0
P
i
i
t)
. This term depends on (i)
contains the additional term i∈{A,B} ω i (Xt ) 1−γαi−a ggi σXσ(X
Y
the weights ω i (Xt ) of each agent, (ii) each agent’s preferences for late or early resolution of
uncertainty (1 − γ i − αi ), (iii) the preference parameters αi , which control agents’ IES, and
(iv) the term
g i0 σX (Xt )
,
gi
σY
which — as we show in the appendix — captures the volatility of
each agent’s consumption-to-total-wealth ratio normalized by the volatility of consumption.
It follows that
κ(Xt )
σY
no longer equals a simple weighted average of individual risk aversions.
To illustrate this point, Figure 3 provides a numerical example where
κ(Xt )
σY
exceeds the risk
aversion of any agent in the economy. In this example, a researcher using a standard,
expected-utility-maximizing framework to infer the risk aversion of the representative agent
(for such an exercise see, e.g., Ait-Sahalia and Lo (2000)) would obtain an estimate exceeding
the maximum risk aversion in the economy.
Since many standard asset-pricing models tend to produce a low market price of risk for
plausible assumptions on agents’ risk aversions, we are particularly interested in determining
i i i0
P
t)
conditions that ensure that the term i∈{A,B} ω i (Xt ) 1−γαi−a ggi σXσ(X
in equation (25) is
Y
positive, so that the market price of risk κ (Xt ) exceeds the Sharpe ratio Γ (Xt ) σY that one
would obtain with heterogeneous, but expected-utility maximizing, agents. The following
proposition addresses this question.
Proposition 3 Consider the same setup as in Proposition 2 with the parameters restricted
to lie in a (arbitrarily large) compact set, and let X̄ denote the stationary mean of Xt . Then
— subject to technical parameter
B
α − αA and sufficiently small
restrictions given in the appendix — for sufficiently small
B
γ − γ A (possibly depending on αB − αA ) the Sharpe
ratio satisfies κ(X̄) ≥ Γ(X̄)σY if either
(i) γ i + αi − 1 > 0 for i ∈ {A, B} and αA < αB , or
(ii) γ i + αi − 1 < 0 for i ∈ {A, B} and αA > αB .
We present the proof of Proposition 3 in the appendix. Here we only give a brief intuition
behind this proposition using a graphical argument. We consider case (i), illustrated in
21
Figure 4; case (ii) can be treated symmetrically. The solid lines depict the paths of expected
(log) consumption for an agent of type A and an agent of type B. According to case (i) of
Proposition 3 , agent A has a lower IES than agent B (αA < αB ). This is reflected in that
agent A’s expected log consumption path in Figure 4 is flatter than that of agent B. Now
suppose that a positive aggregate-consumption shock arrives. The dotted lines describe the
reaction of agents’ log consumption in response to that shock. Since agent A is less risk
averse, her consumption is more strongly exposed to aggregate risk. This is reflected in the
larger vertical distance between the solid and the dotted line for agent A as compared to
agent B.
An additional observation is that the immediate response in either agent’s log consumption is smaller than the response in the long run. Differently put, there is a temporary
increase in each agent’s consumption growth rate, an outcome explained as follows. A positive aggregate shock increases the economic importance of less risk-averse agents (type-A
agents), who, according to the assumptions of case (i), are also the agents with the lower IES.
Since agents of type A become more important in the consumption distribution, equilibrium
asset prices have to adjust to incentivize agents to accept a steeper consumption path.23
This increase in agents’ consumption growth is temporary; it dissipates in the long run as
Xt returns to its long-run mean.
The fact that agents’ consumption reacts more strongly in the long run than in the short
run implies that individual agents’ consumption growth exhibits “long-run risk” in the sense
of Bansal and Yaron (2004). With standard expected utility, agents require compensation
only for the immediate impact of a shock to their consumption. However, with recursive
utility exhibiting a preference for early resolution of uncertainty (as in case (i) of the proposition) they command additional compensation for bearing long-run risk. We stress that
in our setup predictable consumption components arise endogenously and only at the level
of individual agents’ consumption.24 In contrast to Bansal and Yaron (2004), aggregate
23
If the now more economically important type-A agents kept on enjoying the same relatively flat consumption growth path going forward, feasibility would require an excessively steep path for type-B agents,
which they are reluctant to accept at the old prices.
24
See Malloy et al. (2009) for empirical evidence of such predictability.
22
log c B t log c A t log c B t log c A t t
t
t Figure 4: Illustration
of case (i) of Proposition 3.
A
log c
t consumption growth is i.i.d. More importantly,
our framework does not require
stochastic
A
B
log c
log c
t volatility of aggregate consumption in order to produce variation in the market price of risk.
t Finally, we remark that although Proposition 3 applies when preference heterogeneity
is
B
log c
not “too large,” Figure 5 illustrates that the conclusion of the proposition holds even for
non-trivial heterogeneity in preferences.
t
3.3
t
The interest rate (rt )
In Section 2 we discussed why the OLG structure of our economy (absent preference heterogeneity) can lead to a lower level of the interest rate than in the respective infinite-horizon
economy. With preference heterogeneity the interest rate becomes time varying. Motivated
by the stylized facts listed in Campbell and Cochrane (1999), we are particularly interested
whether there exist joint distributions of preference parameters for which the volatility of
discount rates is due almost exclusively to equity-premium, rather than interest-rate, variability.
We can provide an affirmative answer to this question using an approximation (accurate
when σY is not too large) around the steady state.
Proposition 4 Fix D1 > 0 and D2 > 0. Then there exist parameters γ A , γ B , and υ A , as
23
γA=4,γB=10,IESA=1/γA
γA=4,γB=10,IESA=1/γA
0.38
0.36
0.37
0.35
0.36
0.34
0.35
0.33
0.33
IESB=0.85
κ(Xt)
κ(Xt)
0.34
0.32
0.32
IESB=0.05
0.31
0.31
IESB=0.25
0.3
IESB=0.10
0.3
IESB=0.10
0.29
0.05
0.1
0.15
0.2
0.25
0.29
0.3
0.05
Xt
0.1
0.15
0.2
0.25
0.3
Xt
Figure 5: A numerical illustration of Proposition 3. In all cases, we fix agent A’s IES to be equal
to the reciprocal of her risk aversion. The solid lines (in both the left and right panels) depict the
case where agent B is also a CRRA agent. The dashed lines in the left panel depict parameter
combinations corresponding to case (i) of Proposition 3, whereby agent B has preferences for early
resolution of uncertainty and an IES at least as high as agent A. Similarly, the dashed line on the
right hand side panel depicts a case where the more risk averse agent B has preferences for late
resolution of uncertainty and (by implication) a lower IES than agent A.
well as αA and αB possibly depending on σY , so that at the steady-state mean X̄,
κ X̄
= D1 + 1 σY2
σY
0
κ X̄
= D2 + 2 σY2
−
σY
r0 X̄ = 0 + 3 σY2 ,
(28)
(29)
(30)
where the terms i (σY2 ) satisfy limx→0 i (x) = 0 for i ∈ {1, 2} and limx→0
3 (x)
x
= 0. Accord-
ingly, the relative
magnitude
of the volatility of the equity premium to the volatility of the
0
(κ(X̄ )σ (X̄ )) σX (X̄)
interest rate,
, can be made arbitrarily large.
|r0 (X̄ )|σX (X̄)
The main thrust of Proposition 4 is that one can “reverse engineer” a joint distribution
of IES and risk aversion so as to ensure (around the stationary mean of Xt ) that the interest
rate is substantially less volatile than the equity premium. The intuition for this result is
quite simple. As discussed earlier, a positive aggregate shock leads to a decline in the Sharpe
24
ratio. To the first order in σY the magnitude of this decline is determined by the magnitude
of the differences in the risk-aversion coefficients γ of the two agents. However, this decline
in the Sharpe ratio, which reflects the increased importance of relatively more risk-tolerant
agents, also leads to lower aggregate precautionary savings and a higher interest rate. By
choosing the differences in the IES of the two agents judiciously, it is possible to ensure that
the reduction in aggregate precautionary savings is offset by an increase in savings due to
intertemporal-smoothing reasons, leaving the interest rate effectively unchanged. Accordingly, the variation in the interest rate can be made arbitrarily smaller than the variation of
the equity premium.25
From a practical perspective one would also like to know how large these differences in
risk aversion and IES have to be in order to match asset-pricing data. We turn to this
quantitative question in the next section.
4
4.1
Quantitative Implications
Unconditional moments
In this section we illustrate the quantitative implications of the channels highlighted in the
above propositions by determining numerically a population of agents associated with a high
equity premium, volatile returns, and a low and non-volatile interest rate. Specifically, we
choose the same parameters µY , σY , π, ω, and ρ as in Section 2 and the same estimates for
the parametric specification of life-cycle earnings profile (B1 , B2 , δ1 , and δ2 ) as in Figure 2.
(We justify these choices and give detailed references to the data we used in the appendix
— section B). We treat the remaining parameters of the model, namely the risk aversions
and IES’s of agents A and B, as well as the share of type-A agents in the population v A as
free parameters. In the spirit of Proposition 4, we choose these parameters to approximately
match asset-pricing moments and, in particular, the level of the equity premium, the volatility
of returns, the level of the interest rate, and the volatility of the real rate. At a later stage we
25
In addition, it is a simple matter to ensure, through an appropriate parameter choice, that the interestrate level r(X̄) attains any value consistent with finite valuations.
25
Parameter
Risk aversion of type-A agents
Risk aversion of type-B agents
IES of type-A agents
IES of type-B agents
Population share of type-A agents
Equity premium
Volatility of returns
Average (real) interest rate
Volatility of the real interest rate
Data
Baseline
CRRA
1.5
10
5.2 %
18.2%
2.8%
0.92%
1.5
10
0.7
0.05
0.01
5.1 %
17.0 %
1.4%
0.93%
1
γA
1
γB
0.01
2.1 %
8.5 %
2.1 %
0.13 %
Table 1: Unconditional moments for the data, the baseline parametrization and a CRRA
parametrization (restricting the IES to be the inverse of the relative risk aversion). The
data for the average equity premium, the volatility of returns, and the level of the interest
rate are from the long historical sample (1871-2011) available from the website of R. Shiller
(http://www.econ.yale.edu/∼shiller/data/chapt26.xls). The volatility of the “real rate” is based
on the yields of 5-year constant maturity TIPS, as we explain in detail in the text.
also examine whether these parameters, which are chosen to match asset-pricing moments,
also have the correct implications for cross-sectional consumption and wealth data.
We
place an additional requirement, namely that the risk aversion of agent B do not exceed 10,
which is commonly considered the upper bound on plausible values of risk aversion. (See,
e.g., Mehra and Prescott (1985)). This constraint limits the ability of the model to match all
asset-pricing moments exactly, but seems nevertheless important in order to keep the results
comparable with the literature.
Before proceeding, we make two remarks on relating the model’s results to the data. First,
the volatility of the real interest rate is not directly observable in the data. Inferring this
volatility from the difference between the nominal interest rate and realized inflation can be
misleading, since unexpected inflation shocks raise the volatility of the realized real interest
rate above the volatility of the “ex-ante” real interest rate. To circumvent this problem,
we use the yield on the shortest constant-maturity (five years) treasury inflation-protected
securities (TIPS) available from the Federal Reserve. The relatively small volatility (92 basis
points) that we compute this way is a lower bound on the short-rate volatility, since yields
on short-term bonds are more volatile than the respective long-term yields both inside our
26
model and in the data. By aiming to match this number, we make sure to avoid relying on
implausibly volatile real rates.
Second, in order to compare the volatility of our all-equity financed firm to the leveredequity returns observed in the real world, we follow the straightforward approach advocated
by Barro (2006). Barro (2006) proposes to treat levered equity as a zero-net-supply “derivative” security of the positive-supply unlevered equity. The introduction of a zero-net-supply
claim leaves all allocations and prices unchanged, while the Modigliani-Miller theorem implies
that levered equity has the same return as a (constantly rebalanced) replicating portfolio that
is long unlevered equity and short debt. Specifically, given the debt value Bt and the value
t
of levered equity St − Bt , the expected excess rate of return of levered equity is 1 + StB−B
t
times higher than that of unlevered equity. Following Barro (2006), we set
Bt
St −Bt
= 0.5 to
reflect the average historical leverage ratio in NIPA data and report results for levered equity.
The first column of Table 1 gives the empirical values of the moments we are targeting.
The second column gives preference parameters and the resulting model-implied moments.
For comparison purposes, the last column of the table illustrates the CRRA case defined
by the same risk aversions. In the appendix we investigate the model’s robustness to the
choice of parameters; in particular, we show that the results are not affected much when υ A
is constrained to take higher values (Table B.1).
We make two remarks about Table 1. First, the flexibility to choose agents’ IES independent of their risk aversion is important for the model’s quantitative implications. Without this flexibility the model doesn’t perform as well, as illustrated in the third column of
the table. Interestingly, the main reason why the baseline model performs better is not
its higher Sharpe ratio, which only increases from 0.25 to 0.3 when we compare the heterogeneous CRRA case to the recursive-preference case. Rather, the main reason for the
better performance of the baseline model is the higher volatility of returns compared to the
heterogeneous-CRRA case.
In turn, this higher volatility is due to the fact that the variation of the interest rate
reinforces the respective variation of the equity premium, while in the heterogeneous CRRA
27
case it largely offsets it. Specifically, as Figure 6 shows, both the Sharpe ratio and the interest
rate decline with Xt . In other words, the lower compensation required for risk associated
with a positive endowment shock is reinforced by a declining interest rate. Discount rates,
and hence returns, are consequently more volatile.
By contrast, in the CRRA case the interest rate is a hump-shaped function of Xt : As we
discussed earlier (Section 3.3), the sign of the reaction of the interest rate in response to an
increase in Xt depends on the interaction of two effects: a drop in precautionary savings and
a change in savings motivated by intertemporal smoothing. The first effect tends to raise
the interest rate, whereas the second effect tends to lower the interest rate as long as agents
with the higher risk aversion also have higher IES (as is the case with CRRA preferences).
With CRRA preferences the first effect dominates for low values of Xt and the second effect
dominates for higher values of Xt . As a result, over a portion of the state space, the interest
rate counteracts the negative relation between the Sharpe ratio and consumption shocks.
The result is a more muted variation in discount rates and a lower volatility than in our
baseline calibration, where the larger difference in IES between the two agents (compared to
the CRRA case) ensures an interest rate that decreases with Xt .
The fact that in the baseline parametrization the interest rate decreases, while the priceto-dividend ratio increases, with Xt
implies a negative relation between the log-price-
dividend ratio and the real interest rate. In the next section, we show that both in the
data and in the baseline case of the model regressions of the real interest rate on the logprice-dividend ratio yield coefficients that are very small in magnitude and negative.
Our second remark pertains to the type of population required to match asset-pricing
facts. A potential concern with choosing preference parameters to match asset-pricing moments is that such parameters may lead to an implausible concentration of wealth and
consumption in the hands of a small minority, or to excessively volatile cross-sectional consumption and wealth distributions. To address these concerns, we start by noting that an
attractive feature of the OLG structure of the model is that it generates a continuous (rather
than two-point) cross-sectional consumption distribution reflecting the history of aggregate
28
shocks faced by different cohorts of individuals over their life-cycle, and their different savings and portfolio decisions. This adds an important layer of discipline to the model, since
its predictions can be compared to real-world cross-sectional distributions.
Adding support to the model, we find that the same assumptions on the population that
generate the key asset-pricing facts are also helpful in matching a couple of salient features
of real-world consumption and wealth distributions, namely (i) a large Gini coefficient for
wealth inequality and a substantially smaller Gini coefficient for consumption inequality and
(ii) small year-over-year changes in these Gini coefficients. We substantiate these claims
by simulating artificial cross sections of individuals inside the model featuring the same
household numbers and number of cross sections as found in empirical studies.26
Concerning the first feature, which pertains to the level of inequality, we find that the
Gini coefficient of wealth inequality is 0.67 inside the model, while it is 0.9 in the data. At
the same time, the consumption Gini coefficient inside the model is 0.3, which is close to
its data counterpart (0.28). The model can produce a large inequality of wealth and also a
substantially smaller inequality of consumption for two reasons: a) inside the model total
wealth inequality (i.e., wealth inequality measured by combining agents’ financial wealth
and value of human capital) is smaller than pure financial wealth inequality, and b) richer
households tend to have lower consumption-to-total-wealth ratios, i.e., they tend to be better
savers.
The second feature of the data that the model captures is the small variation in the
cross-sectional distributions of consumption and wealth. In contrast to models that need
large yearly movements in these distributions to be consistent with asset-return properties,
our model’s outcome is in line with the data. Specifically, we find that year-over-year changes
in the consumption Gini coefficient27 are about 0.95%, whereas in the data the respective
figure is 0.70%. The reason for the model’s ability to match patterns of asset returns (and
26
The data for the Gini coefficient for financial wealth are from Wolff (2010) and are based on the Survey
of Consumer Finances. The respective data for the consumption of the Gini coefficient are from Cutler and
Katz (1992) and are based on the Consumer Expenditure Survey. We explain in detail how we perform the
simulation of our artificial cross sections in appendix B.
27
Similar conclusions hold for wealth inequality. In the data, year-over-year changes to wealth inequality
are 1.02 %, whereas in the model they are slightly below 1.7%.
29
especially the high variability of returns), while also implying non-volatile cross-sectional
consumption and wealth distributions, is quite simple.
Movements in asset prices are governed exclusively by the current values of Xt and the
aggregate shock. By contrast, an agent’s consumption is affected by the history of past
values of Xt and shocks since her birth. For this reason, our OLG model produces not only
inter-group inequality, but also a large consumption (and wealth) dispersion across different
age cohorts of the same group. This intra-group dispersion reflects the entire path of Xt and
shocks, so that — unlike with asset prices — current shocks affect it relatively little.
We note that the model’s implications for the variability of inequality are likely to be
robust to the introduction of realistic features that, in the interest of parsimony, we left
out. The implications for the level of inequality, on the other hand, would be impacted
by idiosyncratic shocks and bequests, which tend to increase inequality, as well as by taxes
and government redistribution programs, which tend to reduce it. These features could be
added, but would be distractions from the main insights of the paper. For our purposes, it
suffices to note that the model — despite its abstractions — implies properties of inequality
that are not at odds with the data.
We conclude this discussion by noting that the parameters, which we chose to match assetprice properties, are actually in line with the findings of empirical studies utilizing micro-level
data to estimate Euler equations. A stylized fact of such studies is that estimates for the
IES for poorer households tend to be close to zero, while the estimates of the IES for richer
households tend to be substantially higher.28 We note, however, that the precise magnitude
of these estimates can vary widely depending on the instruments chosen to estimate the
parameters, the definition of non-durables and services, the period over which consumption
data are aggregated, etc.
28
These statements hold also conditional on participation. See Vissing-Jorgensen (2002).
30
Horizon
(Years)
Data (Long Sample)
Coefficient
R2
1
-0.13
0.04
3
-0.35
0.09
5
-0.60
0.18
7
-0.75
0.23
Model
Coefficient
R2
-0.09
[-0.29,0.01]
-0.25
[-0.68,0.00]
-0.41
[-1.03,0.01]
-0.54
[-1.27,0.00]
0.03
[0.00,0.10]
0.08
[0.00,0.23]
0.12
[0.00,0.32]
0.16
[0.00,0.41]
Table 2: Long-horizon regressions of excess returns on the log P/D ratio. To account for the
well-documented finite-sample biases driven by the high autocorrelation of the P/D ratio, the
simulated data are based on 1000 independent simulations of 106-year long samples, where the
initial condition for X0 for each of these simulation paths is drawn from the stationary distribution
of Xt . For each of these 106-year long simulated samples, we run predictive regressions of the form
e
= α + β log (Pt /Dt ) , where Rte denotes the gross excess return in period t and h is the
log Rt→t+h
horizon for returns in years. We report the median values for the coefficient β and the R2 of these
regressions, along with the respective [0.025, 0.975] confidence intervals.
4.2
Conditional moments and pathwise comparisons with the data
Besides matching unconditional moments of the data, the model can also match well known
facts pertaining to the predictability of excess returns by such predictive variables as the
price-to-dividend ratio.
Table 2 shows that the model reproduces the empirical evidence on predictability of
excess returns over long horizons. This is not surprising in light of Figure 6: Since dividend
growth is i.i.d. in our model, movements in the price-dividend ratio must be predictive of
changes in discount rates. As Figure 6 shows, interest rate movements in our model are of
much smaller magnitude than excess-return movements. As a result the price-dividend ratio
predicts excess returns quite well.
Table 3 reinforces this point. In this table we use the current log-price-dividend ratio
to predict real riskless returns. Similar to the data, the model produces negative and very
small coefficients of predictability of the riskless rate, while the respective coefficients for
excess returns (Table 2) are an order of magnitude larger. (It is useful to note here that
the negative comovement between the real interest rate and the log price-to-dividend ratio
31
Market price of risk
Equity Premium
0.2
0.5
0.15
0.4
0.1
0.3
0.05
0.2
0
0
0.1
0.2
Xt
0.3
0.1
0
0.4
0.1
Interest rate
0.2
Xt
0.3
0.4
0.3
0.4
Volatility
0.08
0.35
0.3
0.06
0.25
0.04
0.2
0.02
0
0
0.15
0.1
0.2
Xt
0.3
0.1
0
0.4
0.1
0.2
Xt
Figure 6: Equity premium, market price of risk, interest rate, and return volatility for the baseline
parametrization as a function of the consumption share of type-A agents Xt . The range of values
of Xt spans the interval between the bottom 0.5 and the top 99.5 percentiles of the stationary
distribution of Xt . The exact stationary distribution of Xt is given in Figure 7.
should not be interpreted as saying that the real interest rate is “countercyclical”. In our
model all shocks are permanent and hence there is no notion of a “cycle” in the conventional,
Beveridge-Nelson sense nor a notion of an “output gap”.) Finally, we also note that in Table
3 we don’t include comparisons between the R2 of regressions in the data and the model.
The reason is that, unlike regression coefficients, the R2 depends on inflation shocks, which
we do not model.
We conclude with a figure showing that the model can roughly reproduce even more detailed, pathwise properties of excess returns. Specifically, we perform the following exercise.
We take annual consumption data since 1889. We pick the starting value X0 (where year
zero is 1889) to equal the model-implied stationary mean of Xt . For the realized consump-
32
Horizon
Data (Long Sample)
Model
(Years)
1
-0.006
3
-0.039
5
-0.073
7
-0.116
-0.013
[-0.060,0.005]
-0.031
[-0.157,0.012]
-0.047
[-0.248,0.019]
-0.059
[-0.336,0.025]
Table 3: Long-horizon regressions of the real interest rate on the log P/D ratio. To account for
the well-documented finite-sample biases driven by the high autocorrelation of the P/D ratio, the
simulated data are based on 1000 independent simulations of 106-year long samples, where the
initial condition for X0 for each of these simulation paths is drawn from the stationary distribution
of Xt . For each of these 106-year long simulated samples, we run predictive regressions of the form
f
log Rt→t+h
= α + β log (Pt /Dt ) , whereRtf is the gross riskless rate of return at time t and h is the
horizon for returns in years. We report the median values for the coefficient β of these regressions,
along with the respective [0.025, 0.975] confidence intervals.
tion data between 1889 and 2010, we construct the model-implied path of Xt . Since Xt is
the only state variable in the model, we can use knowledge of Xt to construct a series of
model-implied excess returns µt − rt for t = 1889, 1890, and so on. The top panel of Figure
8 compares the model-implied excess returns at time t with the realized excess return in the
data. The bottom panel performs the same exercise averaging returns over rolling eight-year
periods.
Clearly, the model fails to reproduce the data exactly both at the one-year and the
eight-year horizon. This is hardly surprising given the stylized nature of the model. Of
particular interest for our purposes is the behavior of the model for excess returns at eightyear horizons, since averaging helps isolate low-frequency movements in excess returns, which
are more strongly influenced by movements in expected excess returns rather than shocks.29
Focusing on the bottom panel of the figure, we note that the model performs reasonably well
until 1945 and then again from the 1970’s onwards. The model fails to capture the behavior
29
Our results are similar when we consider five- or ten-year averages, or when we isolate business-cycle
frequencies using band-pass filtering.
33
stationary density
7
6
5
4
3
2
1
0
0
0.05
0.1
0.15
Xt
0.2
0.25
0.3
0.35
Figure 7: Stationary density of Xt . The range of values of Xt spans the interval between the
bottom 0.5 and the top 99.5 percentiles of the stationary distribution of Xt .
of average returns between 1945 and 1970. Interestingly, in the data the rolling eight-year
excess return peaks in 1948 and then progressively declines until 1966. In the model excess
returns remain around their long-term average until 1953, while the rapid consumption gains
of the 1950s and 1960s lead to a run-up in excess returns that reaches a peak in 1960, followed
by a decline. One potential explanation for the failure of the model during that period is that
some of the productivity and consumption gains that followed the war were not shocks, but
rather gains anticipated by market participants who understood that productivity “catches
up” rapidly after a war. Accordingly, the model-implied excess returns “lag” the actual data.
5
Concluding remarks
We analyze the asset-pricing implications of preference heterogeneity in a framework combining overlapping generations and recursive utilities. We express the equilibrium in terms
of the solution to a system of ordinary differential equations, and characterize properties of
the solution.
34
Model−implied excess returns and realized excess returns
0.6
Excess returns
0.4
0.2
0
−0.2
−0.4
8−year averages of excess returns
0.2
0.15
0.1
0.05
0
−0.05
Model
Data
−0.1
−0.15
1880
1900
1920
1940
1960
1980
2000
2020
Year
Figure 8: Excess returns according to the model and realized excess returns in the data. To
compute model-implied excess returns, we set the initial value of the consumption share of typeA agents (Xt ) equal to its model-implied stationary mean. Since — according to the model —
consumption growth is i.i.d., we determine consumption innovations by de-meaning first differences
of log consumption growth from 1889 onwards. To account for the different standard deviations
of consumption growth in the pre- and post- World War II sample we divide the de-meaned first
differences of log consumption by the respective standard deviations of the two subsamples. The
resulting series of consumption innovations, together with the initial condition X0 , is used to
compute the model-implied path of Xt . Given Xt , the dotted line depicts the model-implied excess
return for each year (top panel) and the corresponding annualized 8-year excess return (bottom
panel). For comparison, the solid line depicts the data counterparts.
In particular, we show that in the case in which agents have heterogeneous CRRA preferences, the ratio of the market price of risk to the consumption-growth volatility is negatively
correlated with consumption shocks and given by a stationary weighted average of agents’
35
risk aversions. When agents have recursive preferences, however, this ratio is also affected by
the endogenous persistence in individual agents’ consumption growth caused by the optimal
consumption sharing rule. We analyze parameter conditions that lead to a higher market
price of risk. In some cases, the resulting market price of risk may even be higher than in a
world populated exclusively by the agent with the highest risk-aversion coefficient.
Finally, we separate the effects of risk-aversion heterogeneity and IES heterogeneity and
find that risk-aversion heterogeneity is indispensable for the model to have interesting assetpricing implications. However, we also find that the two dimensions of heterogeneity interact
in interesting ways. Importantly, the flexibility to choose IES heterogeneity independently of
risk-aversion heterogeneity allows one to control interest-rate time variation independently
of the variation in the market price of risk.
This flexibility is important for the model’s ability to reproduce asset-pricing facts quantitatively. Interestingly, we find that the type of assumptions on the IES and risk-aversion
heterogeneity required to match asset-pricing facts are also consistent with the salient properties of real-world cross-sectional consumption and wealth distributions. Most importantly,
the interaction of the OLG and preference-heterogeneity features of the model allows for
both a high variability in asset prices and a low variability in cross-sectional measures of
inequality.
36
References
Abel, A. B. (2003). The effects of a baby boom on stock prices and capital accumulation in the
presence of social security. Econometrica 71 (2), 551–578.
Ait-Sahalia, Y. and A. W. Lo (2000). Nonparametric risk management and implied risk aversion.
Journal of Econometrics 94, 9–51.
Backus, D., B. R. Routledge, and S. E. Zin (2008). Who holds risky assets? mimeo, NYU and
CMU.
Bansal, R. and A. Yaron (2004). Risks for the long run: A potential resolution of asset pricing
puzzles. Journal of Finance 59 (4), 1481–1509.
Barro, R. J. (2006). Rare disasters and asset markets in the twentieth century. Quarterly Journal
of Economics 121 (3), 823–66.
Bhamra, H. and R. Uppal (2009). The effect of introducing a non-redundant deerivative on the
volatility of stock-market returns when agents differ in risk aversion. Review of Financial Studies 22, 2303–2330.
Blanchard, O. J. (1985). Debt, deficits, and finite horizons. Journal of Political Economy 93 (2),
223–247.
Branger, N., I. Dumitrescu, V. Ivanova, and C. Schlag (2012). Preference heterogeneity and survival
in long-run risk models. Working paper, University of Frankfurt.
Campbell, J. Y. and J. H. Cochrane (1999). By force of habit: A consumption-based explanation
of aggregate stock market behavior. Journal of Political Economy 107 (2), 205–251.
Campbell, J. Y. and N. G. Mankiw (1989). Consumption, Income, and Interest Rates: Reinterpreting the Time Series Evidence., pp. 185–216. Princeton U and NBER: Cambridge, Mass. and
London:.
Chan, Y. L. and L. Kogan (2002). Catching up with the joneses: Heterogeneous preferences and
the dynamics of asset prices. Journal of Political Economy 110 (6), 1255–1285.
37
Constantinides, G. M., J. B. Donaldson, and R. Mehra (2002). Junior can’t borrow: A new
perspective of the equity premium puzzle. Quarterly Journal of Economics 117 (1), 269–96.
Constantinides, G. M. and D. Duffie (1996). Asset pricing with heterogeneous consumers. Journal
of Political Economy 104, 219–40.
Cutler, D. M. and L. F. Katz (1992). Rising inequality? changes in the distribution of income and
consumption in the 1980’s. American Economic Review 82 (2), 546–551.
Cvitanic, J. and S. Malamud (2010). Relative extinction of heterogeneous agents. The B.E. Journal
of Theoretical Economics 10.
Duffie, D. and L. G. Epstein (1992). Stochastic differential utility. Econometrica 60 (2), 353–394.
Duffie, D., P.-Y. Geoffard, and C. Skiadas (1994). Efficient and equilibrium allocations with stochastic differential utility. Journal of Mathematical Economics 23 (2), 133–146.
Dumas, B. (1989). Two-person dynamic equilibrium in the capital market. Review of Financial
Studies 2 (2), 157–188.
Dumas, B., R. Uppal, and T. Wang (2000). Efficient intertemporal allocations with recursive utility.
Journal of Economic Theory 93, 240–259.
Epstein, L. G. (1987). A simple dynamic general equilibrium model. Journal of Economic Theory 41 (1), 68–95.
Epstein, L. G. and S. E. Zin (1989). Substitution, risk aversion, and the temporal behavior of
consumption and asset returns: A theoretical framework. Econometrica 57 (4), 937–69.
Farmer, R. E. A. (2002). Fiscal policy, equity premia and heterogeneous agents. Working Paper,
UCLA.
Gârleanu, N., L. Kogan, and S. Panageas (2009). The demographics of innovation and asset returns.
NBER Working Paper 15457.
Gentry, W. M. and G. R. Hubbard (2002). Entrepreneurship and household saving. mimeo,
Williams College and Columbia GSB.
38
Gomes, F. and A. Michaelides (2008). Asset pricing with limited risk sharing and heterogeneous
agents. Review of Financial Studies 21 (1), 415–449.
Gomes, J. F., L. Kogan, and M. Yogo (2009). Durability of output and expected stock returns.
Journal of Political Economy 117 (5), 941–986.
Guvenen, F. (2009). A parsimonious macroeconomic model for asset pricing. Econometrica 77 (6),
1711–1740.
Hall, R. E. (1988). Intertemporal substitution in consumption. Journal of Political Economy 96 (2),
339–357.
Heaton, J. and D. J. Lucas (2000). Stock prices and fundamentals. NBER Macroeconomics Annual 14, 213–42.
Hubbard, R. G., J. Skinner, and S. P. Zeldes (1994). The importance of precautionary motives
in explaining individual and aggregate saving. Carnegie-Rochester Conference Series on Public
Policy 40, 59–125.
Kan, R. (1995). Structure of Pareto optima when agents have stochastic recursive preferences.
Journal of Economic Theory 66 (2), 626–631.
Karatzas, I. and S. E. Shreve (1998). Methods of mathematical finance, Volume 39. Springer-Verlag.
Karlin, S. and H. M. Taylor (1981). A Second Course in Stochastic Processes. London, Oxford,
Boston, New York and San Diego: Academic Press.
Kreps, D. M. and E. L. Porteus (1978). Temporal resolution of uncertainty and dynamic choice
theory. Econometrica 46 (1), 185–200.
Longstaff, F. A. and J. Wang (2008). Asset pricing and the credit market. mimeo, UCLA Anderson
and MIT Sloan.
Lucas, Robert E., J. and N. L. Stokey (1984). Optimal growth with many consumers. Journal of
Economic Theory 32 (1), 139–171.
Lucas, R. E. (1978). Asset prices in an exchange economy. Econometrica 46 (6), 1429–1445.
39
Ma, C. (1993). Market equilibrium with heterogenous recursive-utility-maximizing agents. Economic Theory 3 (2), 243–266.
Malloy, C., T. Moskowitz, and A. Vissing-Jorgensen (2009). Long-run stockholder consumption
risk and asset returns. Journal of Finance 64 (6), 2427–2479.
Mehra, R. and E. C. Prescott (1985). The equity premium: A puzzle. Journal of Monetary
Economics 15 (2), 145–161.
Papanikolaou, D. (2010). Investment-specific shocks and asset prices. Working Paper, Kellogg
School of Management.
Piazzesi, M. and M. Schneider (2009). Inflation and the price of real assets. Federal Reserve Bank
of Minneapolis, Staff Report: 423.
Piazzesi, M., M. Schneider, and S. Tuzel (2007). Housing, consumption and asset pricing. Journal
of Financial Economics 83 (3), 531–569.
Santos, T. and P. Veronesi (2006). Labor income and predictable stock returns. Review of Financial
Studies 19 (1), 1–44.
Schroder, M. and C. Skiadas (1999). Optimal consumption and portfolio selection with stochastic
differential utility. Journal of Economic Theory 89 (1), 68–126.
Spear, S. E. and S. Srivastava (1986). Markov rational expectations equilibria in an overlapping
generations model. Journal of Economic Theory 38 (1), 35–62.
Storesletten, K., C. I. Telmer, and A. Yaron (2007). Asset pricing with idiosyncratic risk and
overlapping generations. Review of Economic Dynamics 10 (4), 519–548.
Tuzel, S. (2010). Corporate real estate holdings and the cross-section of stock returns. Review of
Financial Studies 23 (6), 2268–2302.
Vissing-Jorgensen, A. (2002). Limited asset market participation and the elasticity of intertemporal
substitution. Journal of Political Economy 110 (4), 825–853.
40
Wang, J. (1996). The term structure of interest rates in a pure exchange economy with heterogeneous investors. Journal of Financial Economics 41 (1), 75–110.
Weil, P. (1989). The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary
Economics 24 (3), 401–421.
Wolff, E. N. (2010). Recent trends in household wealth in the united states: Rising debt and the
middle-class squeeze - an update to 2007. Working Paper No. 589. Annandale-on-Hudson, NY:
The Levy Economics Institute of Bard College.
Yaari, M. E. (1965). Uncertain lifetime, life insurance, and the theory of the consumer. Review of
Economic Studies 32, 137–50.
Zapatero, F. and C. Xiouros (2010). The representative agent of an economy with external habit
formation and heterogeneous risk aversion. Review of Financial Studies 23, 3017–3047.
41
A
Proofs
We start with the proof of Proposition 2. We then provide the proofs of Proposition 1 and Corollary
1 as special cases.
Proof of Proposition 2. We start by defining the constants
Ξi1 = −
αi + γ i − 1
αi
αi + γ i − 1
ρ+π
i
i
i
i
,
Ξ
=
(1
−
γ
),
Ξ
=
−
,
Ξ
=
−
,
2
4
3
αi
(1 − αi )(1 − γ i )
αi
(1 − αi )(1 − γ i )
where i ∈ A, B. Furthermore, we let
dξt
ξt
ξ˜u
ξ˜t
≡ −rt dt − κt dBt
≡ e−π(u−t)
(A.1)
ξu
.
ξt
(A.2)
Because existing agents have access to a frictionless annuity market and can trade dynamically in
stocks and bonds without constraints, the results in Duffie and Epstein (1992) and Schroder and
Skiadas (1999) imply that the optimal consumption process for agents with recursive preferences
of the form (2) is given by
Ru i
1
ciu,s
fV (w)dw
1−αi t
=
e
cit,s
i
(1 − γ)Vu,s
i
(1 − γ)Vt,s
!Ξi4
ξ˜u
ξ˜t
!
1
αi −1
.
(A.3)
From this point onwards, we proceed by employing a “guess and verify” approach. First, we guess
that both rt and κt are functions of Xt and that Xt is Markov. Later we verify these conjectures.
Under the conjecture that both rt and κt are functions of Xt and that Xt is Markov, the homogeneity of the recursive preferences in Equation (2) implies that there exist a pair of appropriate
functions g i (Xt ), i ∈ {A, B}, such that the time-t value function of an agent of type i born at time
s ≤ t is given by
i
Vt,s
=
fi
W
t,s
1−γ i
1 − γi
g i (Xt )
(1−γ i )(αi −1)
αi
.
(A.4)
f i denotes the total wealth of the agent given by the sum of her financial wealth and the net
W
t,s
R ∞ ξ˜u
i ≡ Wi + E
ft,s
present value of her earnings: W
t t ξ˜ yu,s du. Using (A.4) along with the first order
t,s
t
42
i g i (X ) . Using this last identity inside
ft,s
condition for optimal consumption VW = fc gives cit,s = W
t
(A.4) and re-arranging gives
i
i
Vt,s
i
c1−γ
t,s
− 1−γ
αi .
=
g(X
)
t
1 − γi
(A.5)
Combining Equations (A.5) and (A.3) gives
i
(1 − γ i )Vu,s
i
(1 − γ i )Vt,s
!
1
1−γ i
g(Xu )
g(Xt )
1
αi
=e
1
1−αi
Ru
t
fVi (w)dw
i
(1 − γ)Vu,s
i
(1 − γ)Vt,s
!Ξi4
ξ˜u
ξ˜t
!
1
αi −1
.
(A.6)
Using the definition of f and Equation (A.5), we obtain
fVi (t) = Ξi1 g i (Xt ) + Ξi3
(A.7)
Equation (A.7) implies that fVi (t) is exclusively a function of Xt . Hence Equation (A.6) implies
that
i
(1−γ i )Vu,s
i
(1−γ i )Vt,s
is independent of s. In turn, Equation (A.3) implies that
Motivated by these observations, henceforth we use the simpler notation
of
i
(1−γ i )Vu,s
i
(1−γ i )Vt,s
ciu,s
, respectively.
cit,s
i
(1−γ )Vui
from Equation
(1−γ i )Vti
ciu,s
is independent of s.
cit,s
i
(1−γ i )Vui
and ccui instead
(1−γ i )Vti
t
and
Solving for
(A.6) and applying Ito’s Lemma to the resulting equation
gives
d (1 − γ i )Vui
= µiV du + σVi (1 − γ i )Vui dBu,
(A.8)
where30
σVi
µiV
30
1 g i0
1 − γi
κ
−
σX
t
γi
γ i Ξi2 g i
≡ − 1 − γ i f i (ciu , Vui ).
≡
(A.9)
(A.10)
Equation (A.10) follows from the definition of V , which implies
i
1 − γ i Vt,s
+
Z
t
1 − γ i f i (cu,s , Vu,s ) du = Et
s
Z
s
43
∞
1 − γ i f i (cu,s , Vu,s ) du.
From the definition of Xt we obtain
Z
t
Xt Yt =
−∞
Z
t
=
−∞
ν A πe−π(t−s) cA
t,s ds
ν A πe−π(t−s) cA
s,s e
Rt A
1
f (w)dw
1−αA s v
(1 − γ A )VtA
(1 − γ A )VsA
ΞA
4
ξ˜t
ξ˜s
!
1
αA −1
ds
(A.11)
Applying Ito’s Lemma to both sides of Equation (A.11), using (A.8), equating the diffusion and
drift components on the left- and right-hand side, and simplifying gives
σX
+ σY
Xt
κt
−1
i
= ΞA
4 σV −
(A.12)
αA
= ν A πβtA − πXt +
µX + Xt µY + σX σY
rt − ρ
Xt + nA Xt ,
1 − αA
(A.13)
where
ni ≡
and q i ≡
1
,
αi −1
q i (q i − 1) 2 Ξi4 (Ξi4 − 1) i 2
i
κt +
σV − q i ΞA
4 κt σV
2
2
,
(A.14)
for i ∈ {A, B}. Similarly, applying Ito’s Lemma to both sides of the good-market
clearing equation
Z
t
ν
Yt =
−∞
A
πe−π(t−s) cA
t,s ds
Z
t
+
−∞
ν B πe−π(t−s) cB
t,s ds,
(A.15)
using (A.8), equating the diffusion and drift components on the left- and right-hand sides and
combining with (A.12) and (A.13) leads to Equations (25) and (26). Using Equation (25) inside
(A.12) gives (23), while (24) follows from (A.13). Finally, Equation (27) follows from (A.14) after
simplifying.
i , g i are indeed functions of X and shows how to
The remainder of the proof shows that βt,s
t
t,s
obtain these functions after solving appropriate ordinary differential equations. To this end, we use
the parametric specification G (u) = B1 e−δ1 u + B2 e−δ2 u , and define
φj (Xt ) ≡ Bj ωEt
Z
t
∞
e−(π+δj )(u−t)
ξu Yu
du,
ξt Yt,
so that an agent’s net present value of earnings at birth is given by Ys
44
(A.16)
P
2
j
j=1 φ
(Xs ) . Next notice
that equation (A.16) implies
e−(π+δj )t Yt ξt φj (Xt ) + Bj ω
Z
t
e−(π+δj )u ξu Yu du = Bj ωEt
Z
∞
e−(π+δj )u ξu Yu du.
(A.17)
s
s
Applying Ito’s Lemma to both sides of equation (A.17), setting the drifts equal to each other, and
using the fact that the right hand side of the above equation is a local martingale with respect to
t (so that its drift is equal to zero) results in the differential equation
2 2 j
σX
d φ
dφj
+
(µX + σX (σY − κ)) + φj (µY − r − π − δj − σY κ) + Bj ω,
2 dX 2
dX
0 =
(A.18)
where j = 1, 2. A similar reasoning allows us to obtain an expression for g i (Xt ) , where i ∈ A, B.
Since at each point in time, an agent’s present value of consumption has to equal her total wealth,
we obtain
Z ∞ ˜ i
Z ∞ ˜ i
i
ft,s
W
ξu cu
ξu cu
1
=
=
E
du
=
E
du.
t
t
i
i
i
g (Xt )
ct,s
ξ˜t ct
ξ˜t cit
t
t
(A.19)
Using (A.3) and (A.5) gives
i
ciu
cit
= e
1
γi
Ru
t
(
Ξi1 g(Xw )+Ξi3
) dw
g i (Xu )
g i (Xt )
−Ξi1
γ
ξ˜u
ξ˜t
!− 1i
γ
.
(A.20)
Using (A.20) inside (A.19) and re-arranging implies that
i
−1−
g (Xt )
Ξi1
γi
Z t γ i −1 1 R u
Rt i i
γ i −1
Ξi
1
i
i i
i
− 1i
γi
γi
i 0 (Ξ1 g (Xw )+Ξ3 ) dw
i s (Ξ1 g (Xw )+Ξ3 ) dw i
˜
˜
γ
γ
e
e
ξt
+
g (Xu ) γ du
ξu
s
is a local martingale. Applying Ito’s Lemma and setting the drift of the resulting expression equal
to zero gives
0 =
2
σX

M1i (M1i
− 1)
dg i
dXt
gi
!2
+
d2 g i
dXt
gi

+
dg i
i dXt
M1 i
g
µX − M2i σX κ +
2
2
κ (Xt ) i
Ξi3
i
i
i i
M2 (M2 − 1) − M2 (r (Xt ) + π) − M1 g + i ,
2
γ
where M1i = −1 −
Ξi1
γi
and M2i =
γ i −1
.
γi
45
(A.21)
Concerning boundary conditions for the ODEs (A.18) and (A.21), we note that at the boundaries Xt = 0 and Xt = 1 the value of σX (Xt ) is zero. Accordingly, we require that all terms in
equations (A.18) and (A.21) that contain σX as a multiplicative factor vanish.31 Accordingly, we
are left with the following conditions when Xt = 0 or Xt = 1:
dφj
µX + φj (µY − r − π − δj − σY κ) + Bj ω,
dX
2
i
κ (Xt ) i
Ξi3
i 1 dg
i
i
i i
µX +
0 = M1 i
M2 (M2 − 1) − M2 (r (Xt ) + π) − M1 g + i .
g dXt
2
γ
0 =
(A.22)
(A.23)
i for i ∈ {A, B} and φj for j = 1, 2 are
We conclude the proof by noting that since both gs,t
i
hP
2
j (X ) . The
φ
functions of Xt , so is β i (Xt ), which is by definition equal to β i (Xt ) = g i (Xt )
s
j=1
i and β i are functions of X verifies the conjecture that X is Markovian and that r
fact that gs,t
t
t
t
t
and κt are functions of Xt , which implies that the value functions of agents i ∈ A, B take the form
(A.4).
Lemma 1 The price-to-output ratio is given by
X
2
St
Xt
1 − Xt
π
s (Xt ) ≡
= A
+ B
−
φj (Xt ) ,
Yt
g (Xt ) g (Xt )
π + δj
(A.24)
j=1
while the volatility of returns is given by
σt =
s0 (Xt )
σX (Xt ) + σY .
s (Xt )
(A.25)
Proof of Lemma 1. Using (8), applying Ito’s lemma to compute d e−πs ξs Wsi , integrating, and
using a transversality condition we obtain
i
Wt,s
= Et
Z
t
∞
e−π(u−t)
ξu i
cu,s − yu,s du.
ξt
2
j
(A.26)
2 i
j
i
dφ
dg
2 d φ
2 d g
Specifically, we require that the terms σX
dX 2 , σX dX 2 , σX dX , and σX dX tend to zero as Xt tends
to either zero or one. These conditions ensure that any economic effects at these boundaries (on the
consumption-to-wealth ratio, the present-value-of-income-to-current-output ratio, etc.) of the type of agents
with zero consumption weight at the boundary occurs only though the anticipated (deterministic) births of
new generations of such agents.
31
46
The market-clearing equations for stocks and bonds imply
X Z
St =
t
i
πe−π(t−s) υ i Wt,s
ds.
i∈{A,B} −∞
(A.27)
Substitution of (A.26) into (A.27) gives
t
X Z
St =
πe
i∈{A,B} −∞
t
Z
t
∞
Z
πe−π(t−s) Et
−
∞
Z
υ Et
−π(t−s) i
−∞
ξu
e−π(u−t) ciu,s du
ξt
e−π(u−t)
t
ds
(A.28)
ξu
yu,s du ds.
ξt
We can compute the first term in (A.28) as
X
υ
t
Z
i
−π(t−s)
πe
Z
Et
−∞
i∈{A,B}
∞
t
ξu
e−π(u−t) ciu,s du
ξt
"
ds
#
i
c
ξ
u u,s
=
πe−π(t−s) cit,s Et
υi
e−π(u−t)
du ds
i
ξ
t ct,s
−∞
t
i∈{A,B}
Z t
X
cit,s
Xt
1 − Xt
−π(t−s)
i
πe
=
υ
ds = Yt A
+
g (Xt )
g (Xt ) g B (Xt )
−∞
Z
X
t
Z
∞
(A.29)
i∈{A,B}
Similarly, using (4) and (5) we can compute the second term in (A.28) as
Z
t
−π(t−s)
πe
Z
Et
−∞
∞
e

t
Z
t
= ω
−∞
= Yt ×
πe−π(t−s) Et
= Yt ×
2
X
j=1
∞
t

 
2
X
ξ
u
e−π(u−t) Yu 
Bj e−δj (u−s)  du ds
ξt
−(π+δj )(t−s)
πe
yu,s du ds
j=1
Z
t
∞
Bj ωEt
t
Yu
du ds
ξ t Yt
−(π+δj )(u−t) ξu
e
−∞
2 Z
X
j=1
ξt
Z
t
2 Z
X
j=1
= Yt ×
−π(u−t) ξu
πe−(π+δj )(t−s) φj (Xt ) ds
−∞
π
φj (Xt ) .
π + δj
(A.30)
Combining (A.29) and (A.30) inside (A.28) gives (A.24). Equation (A.25) follows upon applying
Ito’s Lemma to St and matching diffusion terms.
47
Proof of Corollary 1 and Proposition 1.
Proposition 1 is a special case of Proposition
2 given by γ A = γ B .32 Since σX (Xt ) = 0 when γ A = γ B , Lemma 1 implies that σt = σY when
γ A = γ B . Turning to the proof of Proposition 1, the fact that agents have the same preferences
implies that µX = σX = 0 for all Xt , g i (Xt ) and φj (Xt ) for i ∈ {A, B}, j = 1, 2 are constants,
and hence equation (26) becomes (9), while (25) becomes κ = γσY2 . Furthermore equation (A.21)
becomes an algebraic equation with solution
g = π+
ρ
α γ −
r + σY2 ,
1−α 1−α
2
(A.31)
while the present value of an agent’s earnings at birth, divided by Ys , is equal to
Z
ωEs
s
∞
ξu Yu
du = ω
G(u − s)
ξs Ys,
Z
∞
G(u)e
2 −µ )u
−(r+π+γσY
Y
du .
(A.32)
0
Combining (A.31) with (A.32) and the definition of β leads to (10).
Lemma 2 Assume that all agents have the same preferences. Let χ ≡ ρ + π − α µY − γ2 σ 2 and
assume that χ > π, µY − γ
2
σY
2
> 0, and
R∞
χ 0 G(u)e−χu du
1
< R∞
.
ω
π 0 G(u)e−πu du
(A.33)
Then β > 1 and hence the interest rate given by (9) is lower than the respective interest rate in an
economy featuring an infinitely-lived representative agent.
Proof of Lemma 2. Let r ≡ ρ + (1 − α) µY − γ (2 − α)
2
σY
2
denote the interest rate in the economy
featuring an infinitely lived agent and also let r∗ ≡ µY − γσY2 . Note that χ > π is equivalent to
r > r∗ . We first show that condition (A.33) and χ > απ imply, respectively, the two inequalities
σY2
−r
2
σ2
0 < ρ + (1 − α) [µY + π (1 − β (r∗ ))] − γ (2 − α) Y − r∗ .
2
0 > ρ + (1 − α) [µY + π (1 − β (r))] − γ (2 − α)
(A.34)
(A.35)
Since preferences are homogeneous, equation (A.31) holds. Using the definition of r and β (r) on
R∞
the right hand side of (A.34) and simplifying gives 0 > (1 − α) π 1 − χω 0 G(u)e−χu du , which
32
Assuming existence of an equilibrium, the existence of a steady state follows from σX = 0 for all Xt ,
along with µX (0) = πν A β A (0) > 0, µX (1) = −πν B β B (1) < 0, and the intermediate value theorem.
48
is implied by (A.33). Similarly, using the definition of r∗ and β (r∗ ) inside (A.35) and simplifying
h
i
σ2
gives 0 < ρ + π − α(µY + π − γ 2Y ) (1 − ω) , which is implied χ > απ. Given the inequalities
(A.34) and (A.35), the intermediate value theorem implies that there exists a root of equation (9)
in the interval (r∗ , r) . Accordingly, β > 1.
Remark 5 In an influential paper, Weil (1989) pointed out that the standard representative-agent
model cannot account for the low level of the risk-free rate observed in the data. Motivated by the
low estimates of the IES in Hall (1988) and Campbell and Mankiw (1989), Weil’s reasoning was
that such values of the IES would lead to interest rates that are substantially higher than the ones
observed in the data. Weil referred to this observation as the “low risk-free rate puzzle”. Lemma
2 gives sufficient conditions so that the interest rate in our OLG economy is lower than in the
respective infinitely-lived, representative-agent economy. Whether the key condition (A.33) of the
Lemma holds or not depends crucially on the life-cycle path of earnings. The easiest way to see
this is to assume that ωχ > π and restrict attention to the parametric case G(u) = e−δu , so that
condition (A.33) simplifies to δ >
χπ(1−ω)
ωχ−π .
Hence, the interest rate is lower in the overlapping
generations economy as long as the life cycle path of earnings is sufficiently downward-sloping. The
intuition for this finding is that agents who face a downward-sloping path of labor income need to
save for the latter years of their lives. The resulting increased supply of savings lowers the interest
rate. This insight is due to Blanchard (1985), who considered only the deterministic case and
exponential specifications for G (u). Condition (A.33) generalizes the results in Blanchard (1985).
In particular, it allows G (u) to have any shape, potentially even sections where the life-cycle path
of earnings is increasing.
Proof of Proposition 3. We start by fixing arbitrary values αA , αB , and γ, and throughout
we let γ B = γ and γ A = γ B − ε, ε ≥ 0 . For κ (Xt ) given by (25), we let Z (Xt ; ε) ≡ κ (Xt ) −
Γ (Xt ) σY . Clearly
Z X̄; 0 = 0 (by Proposition 1). To prove the proposition, it suffices to show
dZ (X̄;ε) that
≥ 0. Direct differentiation, along with the fact that σX = 0 when ε = 0, gives
dε
ε=0
dZ X̄; ε =
dε
ε=0
X
i∈{A,B}
i
ω X̄
1 − γ i − ai
αi
49
g i0 X̄, 0
×
g i X̄, 0
!
dσX X̄; ε .
dε
ε=0
(A.36)
To determine the sign of
g i0 (X̄,0)
g i (X̄,0)
, we write equation (A.21) — for ε = 0 — as
"
#
2
i0
i
g
(γσ)
Ξ
0 = M1i i µX +
M2i M2i − 1 − M2i (r (Xt ) + π) + 3 − M1i g i .
g
2
γ
(A.37)
Differentiating (A.37) with respect to X and evaluating the result at X = X̄ using µX = 0 yields
g i0 i 0
i
M
µ
−
g
= M2i r0 ,
1
X
gi
(A.38)
where primes denote derivatives with respect to X. Re-arranging (A.38) and using the definitions
of M1i and M2i gives
M2i
g i0
=
gi
M1i
r0
µ0X − g i
=
αi
r0
.
1 − αi µ0X − g i
(A.39)
A similar computation starting from equation (A.18) and using φj X̄ =
Bj ω
r+π+δj +γσ 2 −µY
1
φj0
=−B ω
r0 .
j
0
φj
− µX
φj
yields
(A.40)
The fact that X̄ is a stable steady state33 implies that µ0X ≤ 0, and accordingly equation (A.39)
implies that
1 g i0
α gi
has the opposite sign from r0 . Similarly, equation (A.40) implies that
φ0j
φj
has the
opposite sign34 from r0 .
We next show that if γ A = γ B = γ and αB > αA , then r0 X̄ > 0. We proceed by supposing
the contrary: γ A = γ B = γ and αB > αA , but r0 ≤ 0. Differentiating equation (26) with respect to
Xt and evaluating the resulting expression at Xt = X̄, we obtain
1
r =
Θ X̄
0
1
1
−
B
1−α
1 − αA
A
[r − ρ] − n + n
B
−π
1 X i i 0
v β .
Θ X̄ i
(A.41)
The definition of ni (Xt ) in equation (27) along with γ A = γ B = γ and σX = 0 gives
2
σY
2 − αA
2 − αB κ2 X̄
1
1
n −n =
−
=γ
−
.
A
B
A
B
(1 − α ) (1 − α )
2γ
1−α
1−α
2
A
33
34
B
Note that when ε = 0, the evolution of Xt is deterministic.
Note that Bj and φj have the same sign.
50
(A.42)
Furthermore, noting that β i = g i
P
j=1,2 φ

π
X
vi β
j
and using (A.39) and (A.40) leads to


X
X
1
αi
i
i
g
φ
+
g
φj
= −r0 π
vi 
j
1 − αi g i − µ0X j=1,2
j=1,2
i∈{A,B}



i + X
X
X φ2j
(α
)
 .
vi 
≤ −r0 π
φj + g i
1 − αi j=1,2
B
ω
j
j=1,2
i 0
1
X
i∈{A,B}
Bj ω
φj
− µ0X

(A.43)
i∈{A,B}
Inequality (A.43) follows from the facts that (a) r0 has been assumed to be non-positive, (b) µ0X ≤ 0,
P
is positive and increasing in µ0X .35
and (c) j=1,2 φj Bj ωh1
−µ0X
φj
Let

X
η ≡ π
i∈{A,B}

i )+ X
X φ2j
(α
.
vi 
φj + g i
1 − αi
Bj ω
j=1,2
Let αA = αB = α. Using φj X =
(A.44)
j=1,2
Bj ω
r+π+δj +γσ 2 −µY
and equation (A.31), we obtain that

ρ
α
− 1−α
r + γ2 σY2
ωBj π + 1−α
+
X
X
α
ωB
j
,
= π
+
2
1 − α j=1,2 r + π + δj + γσY2 − µY
(r + π + δj + γσY2 − µY )
j=1,2

η
where r is given in equation (9). We shall assume that
η < 1 − α,
(A.45)
which is the case, for instance, when ω is sufficiently small.36 Using (A.43) and (A.42) inside (A.41)
gives
r
0
≥
1
Θ X
1
1
−
B
1−α
1 − αA
φj (φj )−1 B1
σY2
η
r0 ,
r−ρ+γ
−
2
Θ X
1
r+π+δ1 +σY κ−µY −µ0X >
1
1
2
0
r+π+δ2 +σY κ−µY −µ0X since δ1 < δ2 . Furthermore, since B1 > −B2 , it follows that φ > −φ . Treating µX as
P
P
a variable and differentiating j=1,2 φj (φj )−1 B1 ω−µ0 with respect to µ0X shows that j=1,2 φj (φj )−1 B1 ω−µ0
j
j
X
X
is increasing in µ0X .
36
Note that an implication of Lemma 2 is that the interest rate satisfies r > µY − γσY2 for any value of ω,
i
2
π+ ρ i − α i (r+ γ2 σY
)
1
1−α
1−α
and hence the expressions r+π+δj +γσ2 −µY and
must approach finite limits as ω goes
2 −µ
r+π+δ
+γσ
j
Y
Y
Y
35
To see that
P
j=1,2
0
j ω−µX
is positive, note that
to zero.
51
1
(φ1 )−1 B1 ω−µ0X
(A.46)
=
or
Θ X − η r0 ≥
The term r − ρ + γ
2
σY
2
1
1
−
B
1−α
1 − αA
σY2
r−ρ+γ
.
2
(A.47)
is positive if37
P
Bj
j=1,2 δj +ρ−µ+ γ σ 2 +π
γσY2
2 Y
,
+ π > ω (ρ + π)
µY −
P
Bj
2
j=1,2 δj +π
σY2
ρ + π (1 − α) > α µY − γ
.
2
(A.48a)
(A.48b)
We note that condition (A.48b) is automatically satisfied when α < 0 and µY ≥ γ
2
σY
2
, while
condition (A.48a) holds when ω is small.
When αA − αB is small (but not zero), continuity implies that the right hand side of (A.47)
has the same sign as
1
1−αB
−
1
,
1−αA
which is positive. However, given the supposition that r0
is non-positive and assumption (A.45), the left-hand side of (A.47) is non-positive, which is a
contradiction. We therefore conclude that, when αB > αA , r0 > 0. A symmetric argument implies
i0
that, when αB < αA , r0 < 0. Recalling that r0 and α1i ggi have the same sign, it follows that the
gi0 X̄,0
P
i
i
( )
term i∈{A,B} ω i X̄ 1−γαi−a gi X̄,0 is positive when either condition 1 or condition 2 of the
( )
Lemma holds.
dσX (X̄;ε) , which we want positive. Direct differentiation of (23) gives
Consider now the term
dε
ε=0
dσX X̄; ε dε
X̄
=
ε=0
X̄ 1 − X̄
dΓ(X̄;ε)
dε
h 1−γ−aA gA0
αA
gA
+1
−
1−γ−αB g B0
αB
gB
i
.
(A.49)
+ γB
37
When the economy is populated by a single agent, we can replicate the steps of the proof of Lemma 2
to prove this fact. Specifically, letting r∗∗ = ρ − γ2 σY2 we arrive at the conclusion that
0 < ρ + (1 − α) [µY + π (1 − β (r∗∗ ))] − γ (2 − α)
σY2
− r∗∗ ,
2
as long as condition (A.48a) holds. Similarly, setting r = ρ + (1 − α) (µY + π) − γ (2 − α)
σ2
0 > ρ + (1 − α) µY + π 1 − β r
− γ (2 − α) Y − r.
2
as long as equation (A.48b) holds. Hence there must exist a root in the interval r∗∗ , r .
52
2
σY
2
gives
The numerator on the right hand side of equation (A.49) is positive, and as long as αA and αB
are close enough,
1−γ−aA g A0
αA
gA
−
1−γ−αB g B0
αB
gB
is arbitrarily close to zero, so that the denominator is
also positive.
The proof can now be concluded by invoking uniform continuity on compact sets. Specifically,
the right-hand size of equation (A.47) is weakly positive, for every γ, if αB ≥ αA and αB − αA is
sufficiently small. For αB − αA sufficiently small, the right-hand-side of (A.49) is also positive. It
follows that, for any value γ, (A.36) is positive for αB − αA and ε sufficiently small. Since γ is
restricted to a compact set, the admissible maximum size of αB − αA and ε is strictly positive.
Proof of Proposition 4.
We start by letting σ̃X (Xt ) ≡
σX (Xt )
σY
and γ̃(Xt ) =
κ(Xt )
σY .
Ex-
pressing equations (23)–(26) in terms of σ̃X (Xt ) and γ̃ (Xt ) (rather than σX (Xt ) and κ (Xt )) and
inspecting the resulting equations shows that there are no first-order σY terms in the equations for
r (Xt ) and γ̃ (Xt ). Rather, the lowest (non-zero) order is two. A similar observation applies for the
differential equations (A.18) and (A.21) for φj and g i , respectively.
To establish (28)–(29) it therefore suffices to show that there exist parameters αA , αB , γ A , γ B ,
and υ A such that
X
κ (Xt )
1 − γ i − αi g i0 σX (Xt )
i
= Γ (Xt ) +
ω (Xt )
σY
αi
g i σY
(A.50)
i∈{A,B}
is equal to D1 when σY = 0, while
κ0 (Xt )
σY
is equal to −D2 . To establish these facts, we proceed in
two steps. First we show that, when αA = αB = α and σY = 0, the second term on the right-hand
side of (A.50) is zero, and so is its derivative with respect to Xt . In the second step we show that
there exist γ A , γ B and small enough υ A such that Γ(X̄) = D1 and Γ0 (X̄) = −D2 . In a third step
we address the qualification of equation (30) that limx→0
3 (x)
x
= 0 and its implication.
Step 1. Let αA = αB ≡ α and σY2 = 0. Then Θ(X) is constant and ni (X) = 0, and equation
(26) gives


r (Xt ) = ρ + (1 − α) µY − π 

X
υ i β i (Xt ) − 1 .
(A.51)
i∈{A,B}
The consumption ratio β i (X), in turn, is given by β i (X) = g i (X)(φ1 (X) + φ2 (X)). Note that,
if g i and φj are constant (i.e., independent of Xt ), then so is r. On the other hand, equations (A.21)
53
and (A.18), under the standing assumption σY2 = 0, are satisfied by constant functions g i and φj as
long as r is constant. We conclude that the system of equations (A.18), (A.21), and (A.51) admit
solutions r, g i , and φj that are independent of Xt and therefore g i0 = g i00 = 0. Accordingly, the
second term on the right-hand side of (A.50) and its derivative with respect to Xt are both zero.
Step 2. Now, fixing αA = αB = α, suppose that we want to find γ A and γ B to achieve
Γ X̄ = D1
2
1
1
Γ
X̄
= −D2 .
Γ0 X̄ = −
−
γA γB
(A.52)
(A.53)
With σY = 0, X̄ = υ A is a constant.38 We compute γ A and γ B explicitly by solving the following
linear system in 1/γ A and 1/γ B :
X̄
1 − X̄
+
= D1−1
A
γ
γB
1
1
− B = D1−2 D2 .
A
γ
γ
(A.54)
(A.55)
The solution is
γ B = D1−1 1 − X̄D1−1 D2
(A.56)
γ A = D1−1 + D1−2 D2 (1 − X̄).
(A.57)
We note that γ B is automatically positive, while γ A > 0 as long as X̄ = υ A is small enough.
For future reference we note that steps 1 and 2 (adapted in a straightforward way) continue to
imply equations (28)–(29) not only in the case αA = αB , but more generally when |αA − αB | =
O(σY2 ).
The next step addresses the variability of the interest rate.
38
X̄ = υ A because, at αA = αB and σY = 0, β A = β B and thus equation (24) implies that µX v A = 0.
54
Step 3. Fix now αB = α at the value chosen in Step 1, let ∆α = αA − αB , and consider the
Taylor expansion
r(X; ∆α, σY2 ) = r0 (X; ∆α) + r1 (X; ∆α)σY2 + o(σY2 )
(A.58)
r0 (X; ∆α, σY2 ) = r00 (X; ∆α) + r10 (X; ∆α)σY2 + o(σY2 ).
(A.59)
We want to show that there exists ∆α such that r00 (X̄; ∆α)+r10 (X̄; ∆α)σY2 = 0. We note that —
as established in step 1 — r00 (X̄; 0) = 0 and that
∂
0
∂∆α r0 (X̄; 0)
6= 0 generically. Now, if r10 (X̄; 0) = 0,
then clearly r00 (X̄; ∆α) + r10 (X̄; ∆α)σY2 = 0. If r10 (X̄; 0) 6= 0, the mapping ∆α 7→
r00 (X̄;∆α)
r10 (X̄;∆α)
equals
zero at zero and has non-zero derivative at zero, so its range contains a neighborhood of zero.
Consequently, for σY2 small enough, ∆α exists such that r00 (X̄; ∆α) + r10 (X̄; ∆α)σY2 = 0.
Furthermore, ∆α is in O(σY2 ), which means — in light of our remark at the end of step 2 —
that by choosing ∆α so as to ensure that 3 is of order o(σY2 ), the remainder terms 1 and 2 remain
of O(σY2 ).39
Now the fact that for the parameter choices made above
0
(κ(X̄ )σ (X̄ )) σX (X̄)
as σY
|r0 (X̄ )|σX (X̄)
approaches zero is immediate in light of (28) -(30), along with (A.25).40
B
Details on the calibration
approaches infinity
For all quantitative exercises, we set µY = 0.02 and σY = 0.041, so that time-integrated data from
our model can roughly reproduce the first two moments of annual consumption growth. We note
that, due to time integration, a choice of instantaneous volatility σY = 0.041 corresponds to a
volatility of 0.033 for model-implied, time-integrated, yearly consumption data, consistent with the
long historical sample of Campbell and Cochrane (1999). The parameter π controls the birth-anddeath rate, and we set π = 0.02 so as to approximately match the birth rate (inclusive of the net
immigration rate) in the US population. (Source: U.S. National Center for Health Statistics, Vital
Statistics of the United States, and National Vital Statistics Reports (NVSR), annual data, 19502006.) We note that since the death and birth rate are the same in the model, π = 0.02 implies that
39
Under common regularity conditions, the solutions of differential equations depend smoothly on the
parameters.
40
Specifically, using the observations we made in steps 1-3 imply that for the parameter choices we made
0
σ (X̄) = o(σY ).
55
on average agents live for 50 years after they start making economic decisions. Assuming that this
age is about 20 years in the data, π = 0.02 implies an average lifespan of 70 years. We set ω = 0.92
to match the fact that dividend payments and net interest payments to households are 1 − ω = 0.08
of personal income. (Source: Bureau of Economic Analysis, National income and product accounts,
Table 2.1., annual data, 1929-2010.) To arrive at this number, we combine dividend and net interest
payments by the corporate sector (i.e., we exclude net interest payments from the government and
net interest payments to the rest of the world), in order to capture total flows from the corporate
to the household sector. We note that the choice of 1 − ω = 0.08 is consistent with the gross profit
share of GDP being about 0.3, since the share of output accruing to capital holders is given by
the gross profit share net of the investment share. As a result, in our endowment economy, which
features no investment, it would be misleading to match the gross profit share, since this would not
appropriately deduct investment, which does not constitute “income” for the households. Instead
it seems appropriate to match the parameter ω directly to the fraction of national income that
accrues in the form of dividends and net interest payments from the domestic corporate sector.
For the specification of the life-cycle earnings G(t − s) we use the life-cycle profile of earnings
estimated in Hubbard et al. (1994). We estimate the parameters B1 , δ1 , B2 , and δ2 using non-linear
least squares to project (5) on the life-cycle profile of earnings estimated in Hubbard et al. (1994).
These parameters are B1 = 30.72, B2 = −30.29, δ1 = 0.0525, and δ2 = 0.0611.
To obtain simulated cross sections of households in Section 4, we simulate 1,000 independent
paths of aggregate shocks over 4,000 years starting with X0 at its stationary mean and isolate the
last 300 years of each path to ensure stationarity. Fixing each of the 1,000 paths in turn, we draw
from the population age distribution the ages of 4,000 artificial households. We then repeat this
exercise for T − k, where k are numbers chosen to reflect the number of years between the cross
sections considered in Wolff (2010) and Cutler and Katz (1992) respectively. As a result, we obtain
1,000 independent, simulated versions with the same cross-sectional and time-series characteristics
as Wolff (2010) and Cutler and Katz (1992).
56
Parameter
Risk aversion of type-A agents
Risk aversion of type-B agents
IES of type-A agents
IES of type-B agents
Population share of type-A agents
Equity premium
Volatility of returns
Average (real) interest rate
Volatility of the real interest rate
Data
Baseline
CRRA
υ A =0.08
1.5
10
5.2 %
18.2%
2.8%
0.92%
1.5
10
0.7
0.05
0.01
5.1 %
17.0 %
1.4%
0.93%
1.5
10
0.6
0.03
0.08
3.7 %
16.1 %
2.3 %
0.86%
1
γA
1
γB
0.01
2.1 %
8.5 %
2.1 %
0.13 %
Table B.1: This table repeats the quantitative exercise of Table 1, except that we include an
additional column where we restrict υ A = 0.08 (in addition to γ B ≤ 10) and determine the rest
of the parameters to match (as closely as possible) the asset pricing moments at the bottom of
the table. The choice υ A = 0.08 is based on the fact that entrepreneurs account for 8% of the
population. A potential motivation for matching the fraction of entrepreneurs is that entrepreneurs
are likely to be more risk tolerant. In fact, an extended version of the model can generate this
outcome endogenously. In a previous version of the paper, we included a time-zero occupational
choice for an agent. We modeled an entrepreneur as someone who accepts a multiplicative shock
(with mean 1) on her time-zero endowment of earnings in exchange for a non-pecuniary benefit
of being self-employed. It followed that the relatively more risk-tolerant agents “self-select” into
entrepreneurship. We also note that setting υ A = 0.1 produces similar results to υ A = 0.08.
57